The Database of Integer Sequences, Part 4 

     Part of the On-Line Encyclopedia of Integer Sequences


This is a section of the main database for the On-Line Encyclopedia of Integer Sequences.

For more information see the following pages:
( www.research.att.com/~njas/sequences/ then )

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    home page:          www.research.att.com/~njas/
    


(start)



%I A067458
%S A067458 0,0,0,1,2,0,4,3,2,1,0,1,0,3,0,1,2,7,4,3,0,1,2,0,3,2,0,3,8,3,0,1,2,4,0,
%T A067458 1,6,8,0,5,0,1,2,5,6,0,3,3,5,9,0,1,2,3,4,5,0,5,6,9,0,1,2,4,6,5,10,0,7,
%U A067458 9,0,1,2,5,4,5,8,10,0,9,0,1,2,3,6,5,6,13,10,0,0
%N A067458 Sum of remainders when n is divided by its nonzero digits.
%C A067458 a(n) = 0 for 0 < n 10.
%e A067458 a(14)= 2 as 1 divides 14 and 2 is the remainder obtained when 14 is divided by 4.
%t A067458 Table[Plus @@ Mod[n, Select[IntegerDigits[n], # != 0 &]], {n, 10, 100}]
%Y A067458 Adjacent sequences: A067455 A067456 A067457 this_sequence A067459 A067460 A067461
%Y A067458 Sequence in context: A068773 A133168 A078909 this_sequence A088330 A128263 A095202
%K A067458 base,easy,nonn
%O A067458 10,5
%A A067458 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Feb 07 2002
%E A067458 Edited and extended by Robert G. Wilson v (rgwv(AT)rgwv.com), Feb 11 2002

%I A088330
%S A088330 0,0,0,1,2,0,4,3,2,1,0,1,0,3,0,1,2,7,4,3,0,1,2,0,3,2,0,3,8,3,0,1,2,4,0,
%T A088330 1,6,8,0,5,0,1,2,5,6,0,3,3,5,9,0,1,2,3,4,5,0,5,6,9,0,1,2,4,6,5,10,0,7,9,
%U A088330 0,1,2,5,4,5,8,10,0,9,0,1,2,3,6,5,6,13,10,0,0,3,8,16,10,5,20,14,12,24,0
%N A088330 Sum of the remainders when n is divided by nonzero numbers obtained by deleting one digit. The sum ranges over all the digits.
%C A088330 a((10^n -1)/9) = n. for n > 2. a(1111111 n times ) = a(A00042(n))= n, n > 2.
%e A088330 a(1234) = Rem[1234/123] + Rem[1234/124]+ Rem[1234/134] + Rem[1234/234] = 4+ 118 + 28 + 64 = 214 where Rem [a/b] = the remainder when a is divided by b.
%Y A088330 Cf. A000042.
%Y A088330 Adjacent sequences: A088327 A088328 A088329 this_sequence A088331 A088332 A088333
%Y A088330 Sequence in context: A133168 A078909 A067458 this_sequence A128263 A095202 A093443
%K A088330 base,nonn
%O A088330 10,5
%A A088330 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Oct 01 2003
%E A088330 More terms from Ray Chandler (rayjchandler(AT)sbcglobal.net), Oct 06 2003

%I A128263
%S A128263 1,1,0,1,2,0,4,3,3,2,0,0,2,4,0,1,1,3,4,2,0,0,4,0,1,2,0,4,6,0,4,5,0,1,8,
%T A128263 3,2,4,0,6,6,0,4,0,6,4,0,0,9,1,0,2,6,0,0,12,0,6,12,0,10,4,12,7,4,0,4,1,
%U A128263 0,8,4,9,6,2,0,4,0,0,12,2,9,6,4,0,2,4,0,0,10,6,8,4,0,0,8,0,2,9,0,1,10,0
%V A128263 1,-1,0,-1,-2,0,4,3,-3,2,0,0,-2,-4,0,-1,1,3,-4,2,0,0,4,0,-1,2,0,-4,6,0,4,-5,0,-1,-8,3,
%W A128263 -2,4,0,-6,-6,0,4,0,6,-4,0,0,9,1,0,2,6,0,0,12,0,-6,-12,0,-10,-4,-12,7,4,0,4,-1,0,8,-4,
%X A128263 -9,-6,2,0,4,0,0,12,2,9,6,-4,0,-2,-4,0,0,10,-6,-8,-4,0,0,8,0,2,-9,0,1,-10,0
%N A128263 Coefficients of L-series for elliptic curve "17a4": y^2 +x*y +y= x^3 -x^2 -x or y^2 +x*y -y= x^3 -x^2.
%H A128263 W. Stein, <a href="http://modular.fas.harvard.edu:8080/mfd/space.html?space=[17,2,[]]&search=17">Modular Forms Database</a>.
%F A128263 a(n) is multiplicative with a(17^e) = 1, a(p^e) = a(p)*a(p^(e-1)) -p*a(p^(e-2)) where a(p) = p minus number of points of elliptic curve modulo p.
%F A128263 G.f. is a period 1 Fourier series of a level 17 weight 2 cusp form which satisfies f(-1 / (17 t)) = 17 (t/i)^2 f(t) where q = exp(2 pi i t).
%F A128263 a(9*n) = -3 * a(n). a(9*n+3) = a(9*n+6) = 0.
%e A128263 q - q^2 - q^4 - 2*q^5 + 4*q^7 + 3*q^8 - 3*q^9 + 2*q^10 - 2*q^13 - ...
%o A128263 (PARI) {a(n)= if(n<1, 0, ellak( ellinit([ 1, -1, 1, -1, 0]), n))}
%o A128263 (PARI) {a(n)= local(A, p, e, x, y, a0, a1); if(n<1, 0, A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==17, 1, a0=1; a1=y=-if(p==2, 1, sum(x=0, p-1, kronecker(4*x^3-3*x^2-2*x+1,p))); for(i=2, e, x=y*a1-p*a0; a0=a1; a1=x); a1))))}
%o A128263 (PARI) {a(n)= local(A, q1, q2, q4); if (n<1, 0, n=2*n-1; A=x*O(x^n); q1 = eta(x+A)/ eta(x^17+A); q2 = eta(x^2+A)/ eta(x^34+A); q4 = eta(x^4+A)/ eta(x^68+A); A = eta(x^2+A)^2* eta(x^34+A)^2* (q4-x^2*q1)* ( q2*(q4 +x^2*q1)* (q4^2 -5*x^2*q1*q4 +x^4*q1^2) -q1*q4* (q1*q4*q2^2 +17*x^6) )/ ( 2* q1*q2*q4* (q4^2 -x^2*q1*q4 +x^4*q1^2) ); polcoeff(A, n))}
%Y A128263 Adjacent sequences: A128260 A128261 A128262 this_sequence A128264 A128265 A128266
%Y A128263 Sequence in context: A078909 A067458 A088330 this_sequence A095202 A093443 A099092
%K A128263 sign,mult
%O A128263 1,5
%A A128263 Michael Somos, Feb 21 2007

%I A095202
%S A095202 0,0,2,0,4,3,6,0,8,4,10,8,12,7,14,0,16,8,18,15,20,11,22,15,24,12,26,7,
%T A095202 28,24,30,0,32,16,34,8,36,19,38,15,40,35,42,32,44,23,46,32,48,24,50,39,
%U A095202 52,27,54,48,56,28,58,39,60,31,62,0,64,44,66,16,68,55,70,63,72,36,74,56
%N A095202 Value of largest k such that (n-1) + (n-2) + (n-3) + ... + (n-k) is a multiple of n, or 0 if no such k exists.
%C A095202 Equivalently, largest k < n such that k-th triangular number (A000217(k)) is a multiple of n, or 0 if no such k exists.
%F A095202 a(2n-1) = 2n-2 for all n >= 1; a(2^n) = 0 for all n >= 1.
%o A095202 (PARI) {a(n) = s=0; saved_k=0; k=0; while(k<n-1, k++; s=s+(n-k); if(s%n==0,saved_k=k)); saved_k}
%Y A095202 Cf. A095200, A095201.
%Y A095202 Cf. A000217 (triangular numbers).
%Y A095202 Adjacent sequences: A095199 A095200 A095201 this_sequence A095203 A095204 A095205
%Y A095202 Sequence in context: A067458 A088330 A128263 this_sequence A093443 A099092 A081236
%K A095202 nonn
%O A095202 1,3
%A A095202 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jun 05 2004
%E A095202 Edited by Rick L. Shepherd (rshepherd2(AT)hotmail.com), Jun 08 2004

%I A093443
%S A093443 2,0,4,4,0,0,2,6,3,5,9,6,1,1,5,0,8,4,3,0,3,0,2,0,1,7,9,2,4,5,6,6,0,8,7,
%T A093443 7,3,5,7,8,5,6,4,8,7,6,1,0,5,4,8,4,5,6,1,4,3,7,2,2,4,3,0,5,6,4,0,9,0,1,
%U A093443 1,3,0,1,5,2,3,7,7,1,0,9,9,5,6,7,3,7,2,2,9,3,8,7,3,0,4,9,4,7,7,7,3,0,1
%N A093443 Decimal expansion of (p*e)^(1/3).
%e A093443 2.04400263596115084303020
%Y A093443 Cf. A000796, A001113, A019609.
%Y A093443 Adjacent sequences: A093440 A093441 A093442 this_sequence A093444 A093445 A093446
%Y A093443 Sequence in context: A088330 A128263 A095202 this_sequence A099092 A081236 A103328
%K A093443 cons,nonn
%O A093443 0,1
%A A093443 Mohammad K. Azarian (azarian(AT)evansville.edu), May 13 2004

%I A099092
%S A099092 1,0,2,0,4,4,0,0,16,8,0,0,16,48,16,0,0,0,96,128,32,0,0,0,64,384,320,64,
%T A099092 0,0,0,0,512,1280,768,128,0,0,0,0,256,2560,3840,1792,256,0,0,0,0,0,2560,
%U A099092 10240,10752,4096,512,0,0,0,0,0,1024,15360,35840,28672,9216,1024,0,0,0
%N A099092 Riordan array (1,2+4x).
%C A099092 Row sums are A063727. Diagonal sums are A052907. The Riordan array (1,s+tx) defines T(n,k)=binomial(k,n-k)s^k(t/s)^(n-k). The row sums satisfy a(n)=s*a(n-1)+t*a(n-2) and the diagonal sums satisfy a(n)=s*a(n-2)+t*a(n-3).
%F A099092 Number triangle T(n, k)=binomial(k, n-k)*2^n; columns have g.f. (2x+4x^2)^k.
%e A099092 Rows begin {1}, {0,2}, {0,4,4}, {0,0,16,8}, {0,0,16,48,16}, ...
%Y A099092 Cf. A038210.
%Y A099092 Adjacent sequences: A099089 A099090 A099091 this_sequence A099093 A099094 A099095
%Y A099092 Sequence in context: A128263 A095202 A093443 this_sequence A081236 A103328 A114122
%K A099092 easy,nonn,tabl
%O A099092 0,3
%A A099092 Paul Barry (pbarry(AT)wit.ie), Sep 25 2004

%I A081236
%S A081236 0,0,0,0,2,0,4,4,0,4,8,8,10,8,0,12,14,14,12,16,0,16,20,20,7,25,23,24,24,
%T A081236 23,28,28,32,28,34,32,34,24,38,36,38,37,40,40,36,45,42,44,41,46,36,48,
%U A081236 50,50,40,52,36,57,54,56,48,61,54,60,64,64,64,64,54,68,68,68,66,68,57
%N A081236 Greatest k<n with mu(n-k)=mu(n)=mu(n+k), where mu=A008683 (Moebius function).
%C A081236 a(A081237(n))=0.
%Y A081236 Adjacent sequences: A081233 A081234 A081235 this_sequence A081237 A081238 A081239
%Y A081236 Sequence in context: A095202 A093443 A099092 this_sequence A103328 A114122 A004174
%K A081236 nonn
%O A081236 1,5
%A A081236 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Mar 11 2003

%I A103328
%S A103328 0,2,0,4,4,0,6,20,6,0,8,56,56,8,0,10,120,252,120,10,0,12,220,792,792,
%T A103328 220,12,0,14,364,2002,3432,2002,364,14,0,16,560,4368,11440,11440,4368,
%U A103328 560,16,0,18,816,8568,31824,48620,31824,8568,816,18,0,20,1140,15504
%N A103328 Triangle T(n, k) read by rows: binomial(2n, 2k+1).
%C A103328 A subset of Pascal's triangle A007318 with only even elements.
%D A103328 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 224.
%e A103328 0
%e A103328 2,0
%e A103328 4,4,0
%e A103328 6,20,6,0
%e A103328 8,56,56,8,0
%e A103328 10,120,252,120,10,0
%e A103328 12,220,792,792,220,12,0
%e A103328 14,364,2002,3432,2002,364,14,0
%e A103328 16,560,4368,11440,11440,4368,560,16,0
%Y A103328 Cf. A091042, A086645, A103327.
%Y A103328 Adjacent sequences: A103325 A103326 A103327 this_sequence A103329 A103330 A103331
%Y A103328 Sequence in context: A093443 A099092 A081236 this_sequence A114122 A004174 A049797
%K A103328 nonn,tabl
%O A103328 0,2
%A A103328 Ralf Stephan, Feb 06 2005

%I A114122
%S A114122 1,0,1,2,0,4,4,0,16,0,16,32,0,64,64,0,256,0,256,512,0,1024,1024,0,4096,
%T A114122 0,4096,8192,0,16384,16384,0,65536,0,65536,131072,0,262144,262144,0,
%U A114122 1048576
%N A114122 Expansion of (1+x)^2/(1+2x-4x^3-4x^4).
%C A114122 Binomial transform is A116404.
%F A114122 a(n)=-2a(n-1)+4a(n-3)+4a(n-4)
%Y A114122 Adjacent sequences: A114119 A114120 A114121 this_sequence A114123 A114124 A114125
%Y A114122 Sequence in context: A099092 A081236 A103328 this_sequence A004174 A049797 A116578
%K A114122 easy,nonn
%O A114122 0,4
%A A114122 Paul Barry (pbarry(AT)wit.ie), Feb 07 2006

%I A004174
%S A004174 1,1,2,0,4,4,2,0,12,8,0,16,0,32,16,16,0,80,0,80,32,0,192,0,
%T A004174 320,0,192,64,272,0,1344,0,1120,0,448,128,0,4352,0,7168,0,3584,
%U A004174 0,1024,256,7936,0,39168,0,32256,0,10752,0,2304,512,0,158720,0
%V A004174 1,-1,2,0,-4,4,2,0,-12,8,0,16,0,-32,16,-16,0,80,0,-80,32,0,-192,0,
%W A004174 320,0,-192,64,272,0,-1344,0,1120,0,-448,128,0,4352,0,-7168,0,3584,
%X A004174 0,-1024,256,-7936,0,39168,0,-32256,0,10752,0,-2304,512,0,-158720,0
%N A004174 Triangle of coefficients of Euler polynomials 2^n*E_n(x) (exponents in increasing order).
%D A004174 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 809.
%H A004174 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b004174.txt">Rows n=0..50 of triangle, flattened</a>
%H A004174 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, December 1972 [alternative scanned copy].
%H A004174 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EulerPolynomial.html">MathWorld: Euler Polynomial</a>
%Y A004174 Cf. A099932
%Y A004174 Adjacent sequences: A004171 A004172 A004173 this_sequence A004175 A004176 A004177
%Y A004174 Sequence in context: A081236 A103328 A114122 this_sequence A049797 A116578 A078050
%K A004174 sign,tabl,nice
%O A004174 0,3
%A A004174 njas

%I A049797
%S A049797 0,0,0,2,0,4,4,4,6,14,4,14,20,16,16,30,22,38,32,30,44,64,38,50,68,68,
%T A049797 66,92,66,94,94,96,122,130,90,124,154,158,136,174,148,188,194,172,210,
%U A049797 254,196,228,240,248,258,308,282,302,284
%N A049797 a(n)=Sum{T(n,k): k=2,3,...,n}, array T as in A049800.
%Y A049797 Adjacent sequences: A049794 A049795 A049796 this_sequence A049798 A049799 A049800
%Y A049797 Sequence in context: A103328 A114122 A004174 this_sequence A116578 A078050 A134271
%K A049797 nonn
%O A049797 1,4
%A A049797 Clark Kimberling (ck6(AT)evansville.edu)

%I A116578
%S A116578 2,0,4,4,4,8,0,11,11,16,9,9,25,25,32,0,31,31,55,55,64,28,28,79,79,115,
%T A116578 115,128,0,97,97,181,181,236,236,255,88,88,256,256,392,392,481,481,512,
%U A116578 0,316,316,601,601,828,828,973,973,1024
%N A116578 Integerization of a truncated Pascal root structure with a power of two level pumping.
%C A116578 I used a backward representation of the roots so that the least comes first: the results behaves like an ecomomics or population curve. When taken as Modulo two one ca see a pattern like that of Pascal's triangle in the zeros and ones. The alternating (t-1)^n polynomials are solved as: (t-1)^n=1 and instead of the 2^n coeffiecents, the roots are used for sequence. It is a unique new approach to the problrem of Pascal's triangle.
%F A116578 a(n) = Table[Table[Floor[2^(n - 1)*Abs[x]] /. NSolve[(x - 1)^n - 1 == 0.x][[m]], {m, n, 1, -1}], {n, 1, 10}]
%e A116578 Triangular form of the sequence:
%e A116578 {2}
%e A116578 {0, 4}
%e A116578 {4, 4, 8}
%e A116578 {0, 11, 11, 16}
%e A116578 {9, 9, 25, 25, 32}
%e A116578 {0, 31, 31, 55, 55, 64}
%t A116578 Table[Table[Floor[2^(n - 1)*Abs[x]] /. NSolve[(x - 1)^n - 1 == 0.x][[m]], {m, n, 1, -1}], {n, 1, 10}] Flatten[a]
%Y A116578 Adjacent sequences: A116575 A116576 A116577 this_sequence A116579 A116580 A116581
%Y A116578 Sequence in context: A114122 A004174 A049797 this_sequence A078050 A134271 A094403
%K A116578 nonn,uned,probation,obsc
%O A116578 0,1
%A A116578 Roger L Bagula (rlbagulatftn(AT)yahoo.com), Mar 21 2006

%I A078050
%S A078050 1,2,0,4,4,4,12,4,20,28,12,68,44,92,180,4,364,356,372,1084,340,1828,2508,
%T A078050 1148,6164,3868,8460,16196,724,33116,31668,34564,97900,28772,167028,224572,
%U A078050 109484,558628,339660,777596,1456916,98276,3012108,2815556,3208660,8839772
%V A078050 1,-2,0,4,-4,-4,12,-4,-20,28,12,-68,44,92,-180,-4,364,-356,-372,1084,-340,-1828,2508,
%W A078050 1148,-6164,3868,8460,-16196,-724,33116,-31668,-34564,97900,-28772,-167028,224572,
%X A078050 109484,-558628,339660,777596,-1456916,-98276,3012108,-2815556,-3208660,8839772
%N A078050 Expansion of (1-x)/(1+x+2*x^2).
%Y A078050 Adjacent sequences: A078047 A078048 A078049 this_sequence A078051 A078052 A078053
%Y A078050 Sequence in context: A004174 A049797 A116578 this_sequence A134271 A094403 A129760
%K A078050 sign
%O A078050 0,2
%A A078050 njas, Nov 17 2002

%I A134271
%S A134271 0,1,2,0,4,4,4,12,12,20,36,44,76,116,164
%N A134271 a(n)=a(n-2)+2a(n-3), n grt 3.
%C A134271 Recurrence in A052947.
%F A134271 O.g.f.: 1/2+1/2*(-2*x-5*x^2+1)/(-1+x^2+2*x^3). a(n) = A052947(n-1) + 2*A052947(n-2) - A052947(n-3) . - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 01 2008
%Y A134271 Cf. A078050.
%Y A134271 Adjacent sequences: A134268 A134269 A134270 this_sequence A134272 A134273 A134274
%Y A134271 Sequence in context: A049797 A116578 A078050 this_sequence A094403 A129760 A057377
%K A134271 nonn
%O A134271 0,3
%A A134271 Paul Curtz (bpcrtz(AT)free.fr), Jan 30 2008

%I A094403
%S A094403 1,1,2,0,4,4,5,1,0,4,0,4,0,4,0,0,13,1,6,0,8,20,9,9,1,23,0,8,12,10,26,0,
%T A094403 11,17,29,1,12,20,8,16,3,1,36,0,0,18,19,1,18,26,13,9,10,0,34,32,30,34,
%U A094403 43,1,8,36,8,0,50,60,43,21,25,1,18,0,12,70,25,45,30,40,4,16,80,72,37,1
%N A094403 a(1) = 1; a(n) = (sum of previous terms)^n mod n.
%e A094403 a(4) = 0 because the previous terms 1, 1, 2 sum to 4, and 4^4 mod 4 is 0. a(5) = 4 because the previous terms 1, 1, 2, 0 sum to 4 and 4^5 mod 5 is 4.
%p A094403 L := [1]; s := 1; p := 2; while (nops(L) < 90) do; if 1>0 then; t := (s^p) mod p; L := [op(L),t]; s := s+t; p := p+1; fi; od; L;
%Y A094403 Adjacent sequences: A094400 A094401 A094402 this_sequence A094404 A094405 A094406
%Y A094403 Sequence in context: A116578 A078050 A134271 this_sequence A129760 A057377 A131772
%K A094403 nonn
%O A094403 1,3
%A A094403 Chuck Seggelin (seqfan(AT)plastereddragon.com), Jun 03 2004

%I A129760
%S A129760 0,0,2,0,4,4,6,0,8,8,10,8,12,12,14,0,16,16,18,16,20,20,22,16,24,24,26,
%T A129760 24,28,28,30,0,32,32,34,32,36,36,38,32,40,40,42,40,44,44,46,32,48,48,50,
%U A129760 48,52,52,54,48,56,56,58,56,60,60,62,0,64,64,66,64,68,68,70,64,72,72,74
%N A129760 Bitwise AND of n-1 and n written in base 2.
%F A129760 a(n) = n AND n-1
%e A129760 a(6) = 6 AND 5 = binary 110 AND 101 = binary 100 = 4.
%o A129760 C: int a(int n) { return n & (n-1); }
%Y A129760 Cf. A038712, A086799.
%Y A129760 Adjacent sequences: A129757 A129758 A129759 this_sequence A129761 A129762 A129763
%Y A129760 Sequence in context: A078050 A134271 A094403 this_sequence A057377 A131772 A021493
%K A129760 easy,nonn
%O A129760 1,3
%A A129760 Russ Cox (rsc(AT)swtch.com), May 15 2007

%I A057377
%S A057377 1,0,0,0,2,0,4,4,6,24,24,68,190,192,904,1420,3106,9940,14572,49268,
%T A057377 102886,225004,652940,1301256,3513806,8591792,19326248,52781148,
%U A057377 120709472,306339824,779682608,1852672272,4847112666,11876028924
%N A057377 Low-temperature partition function expansion for square lattice (Potts model, q=3).
%H A057377 I. Jensen, <a href="http://www.research.att.com/~njas/sequences/b057377.txt">Table of n, a(n) for n = 0..71</a> (from link below)
%H A057377 I. Jensen, <a href="http://www.ms.unimelb.edu.au/~iwan/potts/series/sqp3pf.ser">More terms</a>
%Y A057377 Cf. A057374-A057405.
%Y A057377 Adjacent sequences: A057374 A057375 A057376 this_sequence A057378 A057379 A057380
%Y A057377 Sequence in context: A134271 A094403 A129760 this_sequence A131772 A021493 A084247
%K A057377 nonn
%O A057377 0,5
%A A057377 njas, Aug 29 2000

%I A131772
%S A131772 1,0,1,2,0,4,4,8,0,12,20,20,32,0,52,72,104,104,156,0,228,332,436,592,
%T A131772 592,820,0,1152,1588,2180,2772,3592,3592,4744,0,6332,8512,11284,14876,
%U A131772 18468,23212,23212,29544,0,38056,49340,64216,82684,105896,129108,158652
%N A131772 Partial sums (A131771) equal this sequence excluding zeros located at positions {m*(m+1)/2, m>=0}, with a(0)=1.
%e A131772 Partial sums (A131771) begin:
%e A131772 [1,1,2,4,4,8,12,20,20,32,52,72,104,104,156,228,332,436,592,...].
%e A131772 Second partial sums (A131770) begin:
%e A131772 [1,2,4,8,12,20,32,52,72,104,156,228,332,436,592,...].
%o A131772 (PARI)
%Y A131772 Cf. A131770, A131771 (partial sums).
%Y A131772 Adjacent sequences: A131769 A131770 A131771 this_sequence A131773 A131774 A131775
%Y A131772 Sequence in context: A094403 A129760 A057377 this_sequence A021493 A084247 A070692
%K A131772 nonn
%O A131772 0,4
%A A131772 Paul D. Hanna (pauldhanna(AT)juno.com), Jul 14 2007

%I A021493
%S A021493 0,0,2,0,4,4,9,8,9,7,7,5,0,5,1,1,2,4,7,4,4,3,7,6,2,7,8,1,1,8,6,0,9,
%T A021493 4,0,6,9,5,2,9,6,5,2,3,5,1,7,3,8,2,4,1,3,0,8,7,9,3,4,5,6,0,3,2,7,1,
%U A021493 9,8,3,6,4,0,0,8,1,7,9,9,5,9,1,0,0,2,0,4,4,9,8,9,7,7,5,0,5,1,1,2,4
%N A021493 Decimal expansion of 1/489.
%Y A021493 Adjacent sequences: A021490 A021491 A021492 this_sequence A021494 A021495 A021496
%Y A021493 Sequence in context: A129760 A057377 A131772 this_sequence A084247 A070692 A091684
%K A021493 nonn,cons
%O A021493 0,3
%A A021493 njas

%I A084247
%S A084247 1,2,0,4,4,12,20,44,84,172,340,684,1364,2732,5460,10924,21844,43692,
%T A084247 87380,174764,349524,699052,1398100,2796204,5592404,11184812,22369620,
%U A084247 44739244,89478484,178956972,357913940,715827884,1431655764,2863311532
%V A084247 1,2,0,4,-4,12,-20,44,-84,172,-340,684,-1364,2732,-5460,10924,-21844,43692,-87380,
%W A084247 174764,-349524,699052,-1398100,2796204,-5592404,11184812,-22369620,44739244,-89478484,
%X A084247 178956972,-357913940,715827884,-1431655764,2863311532
%N A084247 a(n)=-a(n-1)+2a(n-2), a(0)=1,a(1)=2.
%C A084247 Binomial transform of A084246. a(n+1)=A077925(n)+1.
%F A084247 a(n)=4/3-(-2)^n/3; G.f.: (1+3x)/((1-x)(1+2x)); E.g.f.: (4exp(x)-exp(-2x))/3.
%Y A084247 Cf. A001045.
%Y A084247 Adjacent sequences: A084244 A084245 A084246 this_sequence A084248 A084249 A084250
%Y A084247 Sequence in context: A057377 A131772 A021493 this_sequence A070692 A091684 A100050
%K A084247 easy,sign
%O A084247 0,2
%A A084247 Paul Barry (pbarry(AT)wit.ie), May 23 2003

%I A070692
%S A070692 0,1,2,0,4,5,0,7,8,0,1,2,0,4,5,0,7,8,0,1,2,0,4,5,0,7,8,0,1,2,0,4,5,0,7,
%T A070692 8,0,1,2,0,4,5,0,7,8,0,1,2,0,4,5,0,7,8,0,1,2,0,4,5,0,7,8,0,1,2,0,4,5,0,
%U A070692 7,8,0,1,2,0,4,5,0,7,8,0,1,2,0,4,5,0,7,8,0,1,2,0,4,5,0,7,8,0,1
%N A070692 n^7 mod 9.
%Y A070692 Adjacent sequences: A070689 A070690 A070691 this_sequence A070693 A070694 A070695
%Y A070692 Sequence in context: A131772 A021493 A084247 this_sequence A091684 A100050 A004482
%K A070692 nonn
%O A070692 0,3
%A A070692 njas, May 13 2002

%I A091684
%S A091684 0,1,2,0,4,5,0,7,8,0,10,11,0,13,14,0,16,17,0,19,20,0,22,23,0,25,26,0,28,
%T A091684 29,0,31,32,0,34,35,0,37,38,0,40,41,0,43,44,0,46,47,0,49,50,0,52,53,0,
%U A091684 55,56,0,58,59,0,61,62,0,64,65,0,67,68,0,70,71,0,73,74,0,76,77,0,79,80
%N A091684 Count, setting 3n to zero.
%C A091684 Multiplicative with a(3^e) = 0, a(p^e) = p^e otherwise. Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu) Jun 09, 2005.
%F A091684 a(n)=product{k=0..2, sum{j=1..n, w(3)^(kj) }}, w(3)=e^(2*pi*i/3), i=sqrt(-1). a(n)=2n/3-n*sin(2*pi*n/3+pi/3)/sqrt(3)-n*cos(2*pi*n/3+pi/3)/3.
%F A091684 G.f.: [x(x^4+2x^3+2x+1)]/[(x^2+x+1)^2(x-1)^2]. - R. Stephan, Jan 29 2004
%F A091684 a(n)=n^3 mod 3n; - Paul Barry (pbarry(AT)wit.ie), Apr 13 2005
%Y A091684 Adjacent sequences: A091681 A091682 A091683 this_sequence A091685 A091686 A091687
%Y A091684 Sequence in context: A021493 A084247 A070692 this_sequence A100050 A004482 A111677
%K A091684 nonn,mult
%O A091684 0,3
%A A091684 Paul Barry (pbarry(AT)wit.ie), Jan 28 2004

%I A100050
%S A100050 0,1,2,0,4,5,0,7,8,0,10,11,0,13,14,0,16,17,0,19,20,0,22,23,0,25,26,0,28,29,0,
%T A100050 31,32,0,34,35,0,37,38,0,40,41,0,43,44,0,46,47,0,49,50,0,52,53,0,55,56,0,58,
%U A100050 59,0,61,62,0,64,65,0,67,68,0,70,71,0,73,74,0,76,77,0,79,80,0,82,83,0,85,86,0
%V A100050 0,1,2,0,-4,-5,0,7,8,0,-10,-11,0,13,14,0,-16,-17,0,19,20,0,-22,-23,0,25,26,0,-28,-29,0,
%W A100050 31,32,0,-34,-35,0,37,38,0,-40,-41,0,43,44,0,-46,-47,0,49,50,0,-52,-53,0,55,56,0,-58,
%X A100050 -59,0,61,62,0,-64,-65,0,67,68,0,-70,-71,0,73,74,0,-76,-77,0,79,80,0,-82,-83,0,85,86,0
%N A100050 A Chebyshev transform of n.
%C A100050 A Chebyshev transform of x/(1-x)^2: if A(x) is the g.f. of a sequence, map it to ((1-x^2)/(1+x^2))A(x/(1+x^2)).
%C A100050 Fully multiplicative with a(p) = 0 if p = 3; p otherwise. David W. Wilson (davidwwilson(AT)comcast.net) Jun 10, 2005.
%F A100050 G.f.: x(1-x^2)/(1-x+x^2)^2; a(n)=2a(n-1)-3a(n-2)+2a(n-3)-a(n-4); a(n)=n*sum{k=0..floor(n/2), (-1)^k*binomial(n-k, k)*(n-2k)/(n-k)}.
%Y A100050 Cf. A099837, A099443, A011655, A100047, A100048, A100051, A091684.
%Y A100050 Adjacent sequences: A100047 A100048 A100049 this_sequence A100051 A100052 A100053
%Y A100050 Sequence in context: A084247 A070692 A091684 this_sequence A004482 A111677 A049271
%K A100050 easy,sign,mult
%O A100050 0,3
%A A100050 Paul Barry (pbarry(AT)wit.ie), Oct 31 2004

%I A004482
%S A004482 1,2,0,4,5,3,7,8,6,10,11,9,13,14,12,16,17,15,19,20,18,22,23,21,25,26,
%T A004482 24,28,29,27,31,32,30,34,35,33,37,38,36,40,41,39,43,44,42,46,47,45,49,
%U A004482 50,48,52,53,51,55,56,54,58,59,57,61,62
%N A004482 Tersum n + 1 (answer recorded in base 10).
%C A004482 Sprague-Grundy values for game of Wyt Queens.
%D A004482 E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 76.
%D A004482 A. Dress, A. Flammenkamp and N. Pink, Additive periodicity of the Sprague-Grundy function of certain Nim games, Adv. Appl. Math., 22, p. 249-270 (1999).
%F A004482 Tersum m + n: write m and n in base 3 and add mod 3 with no carries, e.g. 5 + 8 = "21" + "22" = "10" = 1.
%F A004482 Periodic with period 3 and saltus 3: a(n) = 3[ n/3 ] + ((n+1) mod 3).
%F A004482 a(n)= -3 + Sum_{k=0..n}{1/3*(-5*(k mod 3)+4*((k+1) mod 3)+4*((k+2) mod 3)}, with n>=0 - Paolo P. Lava (ppl(AT)spl.at), Dec 03 2007
%Y A004482 This sequence is row 1 of table A004481.
%Y A004482 A061347(n+1) + n.
%Y A004482 Adjacent sequences: A004479 A004480 A004481 this_sequence A004483 A004484 A004485
%Y A004482 Sequence in context: A070692 A091684 A100050 this_sequence A111677 A049271 A004178
%K A004482 nonn,easy,base
%O A004482 0,2
%A A004482 njas
%E A004482 More terms from Erich Friedman (erich.friedman(AT)stetson.edu).

%I A111677
%S A111677 0,2,0,4,5,4,5,0,4,7,7,9,0,8,9,12,10,0,12,14,11,12,0,16,15,19,17,0,23,
%T A111677 18,17,20,0,22,19,22,23,0,24,25,25,26,0,26,27,28,30,0,29,28,25,29,0,28,
%U A111677 28,26,23,0,24,33,33,30,0,28,30,33,26,0,25,34,27,32,0,32,34,35,42,0,33
%N A111677 Array of primes of the type k concatenated with 2n-1 where k < 2n-1. 1---> no prime 13,23 5---> no prime 17,37,47,67 19,29,59,79,89 211,311,811,911 113,313,613,1013,1213 15---> no prime 317,617,... ... Sequence contains the number of terms in the n-th rows.
%C A111677 Conjecture: a(n)=0 iff n== 3 (mod 5). [Corrected by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 20 2007]
%C A111677 Subsidiary sequences: (1) First occurrence of n in A111677. There are numbers like 3 which probabely do not occur in this sequence, let a(3) = -1. (2) Terms that do not occur in A111677.
%e A111677 For 2n-1 = 9, we have primes 19,29,59,79 and 89. Hence a(5) = 5.
%p A111677 cat2 := proc(n,m) n*10^(max(1,ilog10(m)+1))+m ; end: A111677 := proc(nrow) local town1,k,a ; town1 := 2*nrow-1 ; a := [] ; for k from 1 to town1-1 do if isprime(cat2(k,town1)) then a := [op(a),cat2(k,town1)] ; fi ; od; RETURN(nops(a)) ; end: seq(A111677(nrow),nrow=1..80) ; - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 20 2007
%Y A111677 Cf. A111676.
%Y A111677 Adjacent sequences: A111674 A111675 A111676 this_sequence A111678 A111679 A111680
%Y A111677 Sequence in context: A091684 A100050 A004482 this_sequence A049271 A004178 A068333
%K A111677 base,nonn
%O A111677 1,2
%A A111677 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Aug 16 2005
%E A111677 More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 20 2007

%I A049271
%S A049271 1,2,0,4,5,6,4,8,9,10,11,0,1,2,3,4,5,6,4,8,9,6,11,12,13,14,0,4,8,12,4,
%T A049271 20,21,6,15,24,4,6,3,28,17,30,4,32,33,34,35,0,1,2,3,4,5,6,7,8,9,10,11,
%U A049271 12,4,2,15,16,17,6,4,20,21,22,11,24,25,26,0,4,8,12,4,20,24,6,15,36,37
%N A049271 Smallest nonnegative value taken on by nx^2 - 12y^2 for an infinite number of integer pairs (x, y).
%Y A049271 Adjacent sequences: A049268 A049269 A049270 this_sequence A049272 A049273 A049274
%Y A049271 Sequence in context: A100050 A004482 A111677 this_sequence A004178 A068333 A121451
%K A049271 nonn
%O A049271 1,2
%A A049271 David W. Wilson (davidwwilson(AT)comcast.net)

%I A004178
%S A004178 0,1,2,0,4,5,6,7,8,9,10,11,12,1,14,15,16,17,18,19,20,21,22,2,24,25,
%T A004178 26,27,28,29,0,1,2,0,4,5,6,7,8,9,40,41,42,4,44,45,46,47,48,49,50,51,
%U A004178 52,5,54,55,56,57,58,59,60,61,62,6,64,65,66,67,68,69,70,71,72,7,74
%N A004178 Omit 3's from n.
%Y A004178 Adjacent sequences: A004175 A004176 A004177 this_sequence A004179 A004180 A004181
%Y A004178 Sequence in context: A004482 A111677 A049271 this_sequence A068333 A121451 A096984
%K A004178 nonn,base
%O A004178 0,3
%A A004178 njas

%I A068333
%S A068333 0,1,2,0,4,5,6,14,0,27,10,44,12,65,28,0,16,357,18,152,80,189,22,2300,0,
%T A068333 275,156,972,28,2639,30,1736,256,495,68,0,36,629,380,12636,40,8569,42,
%U A068333 6020,2112,945,46,215072,0,5635,700,11016,52,59625
%N A068333 Product(n/k - k) where the product is over the divisors k of n and where 1 <= k <= sqrt(n).
%e A068333 a(8) = (8 - 1) (4 - 2) = 14 because 1 and 2 are the divisors of 8 which are <= sqrt(8).
%Y A068333 Adjacent sequences: A068330 A068331 A068332 this_sequence A068334 A068335 A068336
%Y A068333 Sequence in context: A111677 A049271 A004178 this_sequence A121451 A096984 A104601
%K A068333 nonn
%O A068333 1,3
%A A068333 Leroy Quet (qq-quet(AT)mindspring.com), Feb 27 2002

%I A121451
%S A121451 0,2,0,4,5,8,10,16,20,32,40,64,80,128,160,256,320,512,640,1024,1280,
%T A121451 2048,2560,4096,5120,8192,10240,16384,20480,32768,40960,65536,81920,
%U A121451 131072,163840,262144,327680,524288,655360,1048576
%N A121451 Maximum product over partitions into parts of the form 3k+2.
%C A121451 With the exception of the first three terms of this sequence and the first two terms of A094958, these two sequences appear to be identical.
%F A121451 Conjecture. a(1)=a(3)=0, otherwise a(n)=2^(n/2) if n is even and a(n)=5*2^((n-5)/2) if n is odd. (This jas been verified for up to n=40.)
%e A121451 The only partition of 7 into parts of the form 3k+2 is {5,2}, so the maximum product is a(7)=10.
%Y A121451 Cf. A000792, A034893, A094958.
%Y A121451 Adjacent sequences: A121448 A121449 A121450 this_sequence A121452 A121453 A121454
%Y A121451 Sequence in context: A049271 A004178 A068333 this_sequence A096984 A104601 A133144
%K A121451 nonn
%O A121451 1,2
%A A121451 John W. Layman (layman(AT)math.vt.edu), Apr 26 2007

%I A096984
%S A096984 2,0,4,5,96,427,6448,56961,892720,11905091,211153944
%N A096984 Another version of A005512, which is the main entry for this sequence.
%Y A096984 Adjacent sequences: A096981 A096982 A096983 this_sequence A096985 A096986 A096987
%Y A096984 Sequence in context: A004178 A068333 A121451 this_sequence A104601 A133144 A098123
%K A096984 dead
%O A096984 2,1

%I A104601
%S A104601 1,0,2,0,4,6,0,1,45,24,0,0,90,432,120,0,0,78,2248,4200,720,0,0,36,5776,
%T A104601 43000,43200,5040,0,0,9,9066,222925,755100,476280,40320,0,0,1,9696,
%U A104601 727375,6700500,13003620,5644800,362880,0,0,0,7480,1674840
%N A104601 Triangle T(r,n) read by rows: number of n X n (0,1)-matrices with exactly r entries equal to 1, and no zero row or columns.
%H A104601 M. Maia and M. Mendez, <a href="http://arXiv.org/abs/math.CO/0503436">On the arithmetic product of combinatorial species</a>
%F A104601 T(r, n)=Sum{l>=r, Sum{d|l, (-1)^(2n-d-l/d)*C(n, d)*C(n, l/d)*C(l, r) }}.
%F A104601 E.g.f.: Sum(((1+x)^n-1)^n*exp((1-(1+x)^n)*y)*y^n/n!,n=0..infinity). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Feb 24 2008
%e A104601 1
%e A104601 0,2
%e A104601 0,4,6
%e A104601 0,1,45,24
%e A104601 0,0,90,432,120
%e A104601 0,0,78,2248,4200,720
%e A104601 0,0,36,5776,43000,43200,5040
%e A104601 0,0,9,9066,222925,755100,476280,40320
%e A104601 0,0,1,9696,727375,6700500,13003620,5644800,362880
%e A104601 0,0,0,7480,1674840,37638036,179494350,226262400,71850240,3628800
%Y A104601 Right-edge diagonals include A000142, A055602, A055603. Row sums are in A104602.
%Y A104601 Column sums are in A048291. The triangle read by columns = A055599.
%Y A104601 Adjacent sequences: A104598 A104599 A104600 this_sequence A104602 A104603 A104604
%Y A104601 Sequence in context: A068333 A121451 A096984 this_sequence A133144 A098123 A066659
%K A104601 nonn,tabl
%O A104601 1,3
%A A104601 Ralf Stephan, Mar 27 2005

%I A133144
%S A133144 0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,0,4,6,3,2,1,1,1,7,
%T A133144 3,2,3,6,3,7,1,1,2,2,4,3,3,3,2,5,1,1,2,5,2,2,6,5,4,2,1,1,3,7,2,4,4,
%U A133144 5,4,5,1,1,4,4,5,2,3,6,4,3,1,1,4,5,3,5,5,3,4,5,1,1,3,9,4,6,2,2,5,3
%N A133144 Start with n and repeatedly apply the powerback map of A133048. Sequence gives number of steps to the point where the next number would be one that has appeared before.
%C A133144 It is conjectured that every number eventually reaches a fixed point (see A131571) or the cycle of length 2 given by (175 <-> 78125).
%e A133144 n, a(n), trajectory
%e A133144 22, 1, [22, 4]
%e A133144 23, 1, [23, 9]
%e A133144 24, 2, [24, 16, 6]
%e A133144 25, 0, [25]
%e A133144 26, 4, [26, 36, 216, 12, 2]
%e A133144 27, 6, [27, 49, 6561, 15625, 194400, 2304, 9]
%e A133144 28, 3, [28, 64, 4096, 0]
%e A133144 29, 2, [29, 81, 1]
%e A133144 30, 1, [30, 3]
%e A133144 31, 1, [31, 1]
%e A133144 32, 1, [32, 8]
%e A133144 33, 7, [33, 27, 49, 6561, 15625, 194400, 2304, 9]
%e A133144 34, 3, [34, 64, 4096, 0]
%e A133144 35, 2, [35, 125, 25]
%e A133144 36, 3, [36, 216, 12, 2]
%e A133144 37, 6, [37, 343, 243, 162, 64, 4096, 0]
%e A133144 38, 3, [38, 512, 10, 1]
%e A133144 39, 7, [39, 729, 567, 588245, 5242880000, 8589934592, 105911076180375000000000, 0]
%Y A133144 Adjacent sequences: A133141 A133142 A133143 this_sequence A133145 A133146 A133147
%Y A133144 Sequence in context: A121451 A096984 A104601 this_sequence A098123 A066659 A085623
%K A133144 nonn,base
%O A133144 0,24
%A A133144 J. H. Conway and njas, Jan 01 2008

%I A098123
%S A098123 1,0,0,2,0,4,6,6,24,28,60,130,190,432,770,1386,2856,5056,9828,18918,
%T A098123 34908,68132,128502,244090,470646,890628,1709136,3271866,6238986,
%U A098123 11986288,22925630,43932906,84349336,161625288,310404768,596009494
%N A098123 Number of compositions of n with equal number of even and odd parts.
%F A098123 a(n) = Sum_{k=floor(n/3)..floor(n/2)} binomial(2*n-4*k, n-2*k)*binomial(n-1-k, 2*n-4*k-1).
%Y A098123 Cf. A045931.
%Y A098123 Adjacent sequences: A098120 A098121 A098122 this_sequence A098124 A098125 A098126
%Y A098123 Sequence in context: A096984 A104601 A133144 this_sequence A066659 A085623 A002885
%K A098123 easy,nonn
%O A098123 0,4
%A A098123 Vladeta Jovovic (vladeta(AT)Eunet.yu), Sep 24 2004

%I A066659
%S A066659 2,0,4,6,8,0,9,10,14,12,22,0,21,18,16,20,32,0,27,24,26,0,46,30,33,2,8,
%T A066659 38,36,58,0,62,34,44,40,39,42,57,54,45,48,55,0,49,50,52,0,94,60,86,66,
%U A066659 64,56,106,0,75,70,63,0,118,0,77,0,74,68,104,0,134,80,92,72,142,78,91
%N A066659 a(n) = least k > n such that EulerPhi(k) = EulerPhi(n), if such k exists; = 0 otherwise.
%Y A066659 Cf. A000010.
%Y A066659 Adjacent sequences: A066656 A066657 A066658 this_sequence A066660 A066661 A066662
%Y A066659 Sequence in context: A104601 A133144 A098123 this_sequence A085623 A002885 A011121
%K A066659 nonn
%O A066659 1,1
%A A066659 Joseph L. Pe (joseph_l_pe(AT)hotmail.com), Jan 10 2002
%E A066659 More terms from Vladeta Jovovic (vladeta(AT)Eunet.yu), Jan 12 2002

%I A085623
%S A085623 0,2,0,4,6,10,4,12,18,4,14,18,20,16,30,32,30,20,28,34,32,40,46,54,46,48,
%T A085623 64,62,66,40,68,66,72,90,68,70,84,92,90,100,90,80,98,102,88,88,108,108,
%U A085623 106,126,116,126,112,134,136,150,116,142,146,144,146,136,156,158,178
%N A085623 Let p = n-th odd prime; a(n) = number of pairs (i,j) with 0 < i < p, 0 < j < p such that ij == 1 mod p and i and j have opposite parity.
%D A085623 R. K. Guy, Unsolved Problems in Number Theory, F12.
%D A085623 Yuan Yi and Zhang Wen-Peng, On the generalization of a problem of D. H. Lehmer, Kyushu J. Math., 56 (2002) 235-241; MR 2003g:11112.
%t A085623 f[n_] := Length[ Select[ Mod[ Flatten[ Table[i*j, {j, 2, n - 1}, {i, j - 1, 1, -2}], 1], n], # == 1 & ]]; 2Table[ f[ Prime[n]], {n, 2, 70}]
%Y A085623 Adjacent sequences: A085620 A085621 A085622 this_sequence A085624 A085625 A085626
%Y A085623 Sequence in context: A133144 A098123 A066659 this_sequence A002885 A011121 A117902
%K A085623 nonn,easy
%O A085623 2,2
%A A085623 njas, based on a suggestion of R. K. Guy (rkg(AT)cpsc.ucalgary.ca), Jul 11 2003
%E A085623 Extended by Vladeta Jovovic (vladeta(AT)Eunet.yu) and Robert G. Wilson v (rgwv(AT)rgwv.com), Jul 12 2003

%I A002885 M0032 N0393
%S A002885 1,1,0,1,0,0,1,2,0,4,7,0,12,8,0,80,84,0,820,798,0,9508,11616,0,157340,
%T A002885 139828,0,3027456,2353310
%N A002885 Number of cyclic Steiner triple systems of order 2n+1.
%D A002885 J. Doyen, Problem 30, p. 504 of R. K. Guy et al., editors, Combinatorial Structures and Their Applications (Proceedings Calgary Conference Jun 1969}), Gordon and Breach, NY, 1970.
%D A002885 C. J. Colbourn and A. Rosa: Triple Systems, Clarendon Press (Oxford) 1999
%H A002885 <a href="http://www.research.att.com/~njas/sequences/Sindx_St.html#Steiner">Index entries for sequences related to Steiner systems</a>
%Y A002885 Adjacent sequences: A002882 A002883 A002884 this_sequence A002886 A002887 A002888
%Y A002885 Sequence in context: A098123 A066659 A085623 this_sequence A011121 A117902 A021087
%K A002885 nonn
%O A002885 0,8
%A A002885 njas
%E A002885 More terms from Michael Steyer (m.steyer(AT)osram.de), Jan 27 2005

%I A011121
%S A011121 2,0,4,7,6,7,2,5,1,1,0,7,9,2,1,9,2,9,6,2,1,2,8,3,7,3,5,6,3,2,8,6,2,
%T A011121 1,8,7,5,4,9,6,2,1,9,1,8,5,1,9,6,6,9,0,2,1,1,9,5,5,8,2,1,6,3,1,8,6,
%U A011121 1,5,0,8,6,5,2,4,2,5,8,9,2,1,3,3,8,7,0,1,8,2,1,2,7,3,3,9,9,4,6,4,8
%N A011121 Decimal expansion of 5th root of 36.
%Y A011121 Adjacent sequences: A011118 A011119 A011120 this_sequence A011122 A011123 A011124
%Y A011121 Sequence in context: A066659 A085623 A002885 this_sequence A117902 A021087 A120558
%K A011121 nonn,cons
%O A011121 1,1
%A A011121 njas

%I A117902
%S A117902 1,0,1,2,0,4,8,0,16,32,0,64,128,0,256,512,0,1024,2048,0,4096,8192,0,16384,
%T A117902 32768,0,65536,131072,0,262144,524288,0,1048576,2097152,0,4194304,8388608,0,
%U A117902 16777216,33554432,0,67108864,134217728,0,268435456,536870912,0,1073741824
%V A117902 1,0,-1,2,0,-4,8,0,-16,32,0,-64,128,0,-256,512,0,-1024,2048,0,-4096,8192,0,-16384,
%W A117902 32768,0,-65536,131072,0,-262144,524288,0,-1048576,2097152,0,-4194304,8388608,0,
%X A117902 -16777216,33554432,0,-67108864,134217728,0,-268435456,536870912,0,-1073741824
%N A117902 Expansion of (1-x^2-2x^3)/(1-4x^3).
%C A117902 Row sums of number triangle A117901.
%F A117902 a(n)=0^n/2-2^(2n/3)(cos(2*pi*n/3+pi/3)/6+sqrt(3)*sin(2*pi*n/3+pi/3)/6 -(2^(2/3)/12+2/3)cos(2*pi*n/3)-432^(1/6)*sin(2*pi*n/3)/12+2^(2/3)/12-1/6)
%Y A117902 Adjacent sequences: A117899 A117900 A117901 this_sequence A117903 A117904 A117905
%Y A117902 Sequence in context: A085623 A002885 A011121 this_sequence A021087 A120558 A120554
%K A117902 easy,sign
%O A117902 0,4
%A A117902 Paul Barry (pbarry(AT)wit.ie), Apr 01 2006

%I A021087
%S A021087 0,1,2,0,4,8,1,9,2,7,7,1,0,8,4,3,3,7,3,4,9,3,9,7,5,9,0,3,6,1,4,4,5,
%T A021087 7,8,3,1,3,2,5,3,0,1,2,0,4,8,1,9,2,7,7,1,0,8,4,3,3,7,3,4,9,3,9,7,5,
%U A021087 9,0,3,6,1,4,4,5,7,8,3,1,3,2,5,3,0,1,2,0,4,8,1,9,2,7,7,1,0,8,4,3,3
%N A021087 Decimal expansion of 1/83.
%Y A021087 Adjacent sequences: A021084 A021085 A021086 this_sequence A021088 A021089 A021090
%Y A021087 Sequence in context: A002885 A011121 A117902 this_sequence A120558 A120554 A120710
%K A021087 nonn,cons
%O A021087 0,3
%A A021087 njas

%I A120558
%S A120558 2,0,4,8,2,24,20,18,88,40,108,246,86,408,612,242,1350,1224,968,3540,
%T A120558 2840,2884,9176,4812,11314,17604,16484,22376,46602,26128,88204,
%U A120558 43816,153144,73004,275012,103868,604014,198132,1533348,605098
%V A120558 2,0,4,8,2,24,20,18,88,40,108,246,86,408,612,242,1350,1224,968,3540,
%W A120558 2840,2884,9176,4812,11314,17604,16484,22376,46602,26128,88204,
%X A120558 43816,153144,73004,275012,103868,604014,-198132,1533348,-605098
%N A120558 Site series for first parallel moment of 4.8 (bathroom tile) lattice.
%H A120558 I. Jensen, <a href="http://www.research.att.com/~njas/sequences/b120558.txt">Table of n, a(n) for n = 0..238</a> [from link below]
%H A120558 I. Jensen, <a href="http://www.ms.unimelb.edu.au/~iwan/dirperc/series/4_8site_t1.ser">More terms</a>
%Y A120558 Adjacent sequences: A120555 A120556 A120557 this_sequence A120559 A120560 A120561
%Y A120558 Sequence in context: A011121 A117902 A021087 this_sequence A120554 A120710 A115780
%K A120558 sign
%O A120558 1,1
%A A120558 njas, Aug 09 2006

%I A120554
%S A120554 2,0,4,8,10,16,44,48,82,156,236,300,614,820,1178,1792,3330,3508,
%T A120554 6598,8960,13716,15744,36688,31868,61454,75472,150812,96100,366904,
%U A120554 217988,594880,386124,1728530,113384,3694726,401424,6743452
%N A120554 Bond series for first parallel moment of 4.8 (bathroom tile) lattice.
%H A120554 I. Jensen, <a href="http://www.research.att.com/~njas/sequences/b120554.txt">Table of n, a(n) for n = 0..254</a> [from link below]
%H A120554 I. Jensen, <a href="http://www.ms.unimelb.edu.au/~iwan/dirperc/series/4_8bond_t1.ser">More terms</a>
%Y A120554 Adjacent sequences: A120551 A120552 A120553 this_sequence A120555 A120556 A120557
%Y A120554 Sequence in context: A117902 A021087 A120558 this_sequence A120710 A115780 A101189
%K A120554 nonn
%O A120554 1,1
%A A120554 njas, Aug 09 2006

%I A120710
%S A120710 0,0,0,2,0,4,8,14,0,8,16,26,32,44,56,70,0,16,32,50,64,84,104,126,128,
%T A120710 152,176,202,224,252,280,310,0,32,64,98,128,164,200,238,256,296,336,378,
%U A120710 416,460,504,550,512,560,608,658,704,756,808,862,896,952,1008,1066,1120
%N A120710 A GF(2) polynomial analog of triangular numbers.
%C A120710 The k-th bit in a(n) is one just if there are an odd number of pairs of distinct one bits i#j in n such that i+j=k. GF(2) polynomial ("XOR numbral") multiplication can be implemented as A048720(i,j) = A000695(i AND j) XOR a(i AND j) XOR a(i IOR j) XOR a(i AND NOT j) XOR a(NOT i AND j), analogously to ordinary multiplication (A003991) ij = tri(i+j)-tri(i)-tri(j) via triangular numbers (A000217).
%D A120710 Posting by Richard Schroeppel (rschroe(AT)sandia.gov) to math-fun mailing list, Jun 26 2006.
%F A120710 a(0)=0; a(n + 2^k) = a(n) XOR (n * 2^k), 0<=n<2^k.
%e A120710 a(15)=54 because 15=2^0+2^1+2^2+2^3, the four one-bits giving six distinct pairs 01 02 03 12 13 23, which sum to 1 2 3 3 4 5, of which 1 2 4 and 5 occur oddly, yielding 2^1+2^2+2^4+2^5=54.
%Y A120710 Cf. A048720, A000695, A003991, A000217.
%Y A120710 Adjacent sequences: A120707 A120708 A120709 this_sequence A120711 A120712 A120713
%Y A120710 Sequence in context: A021087 A120558 A120554 this_sequence A115780 A101189 A070015
%K A120710 base,easy,nonn
%O A120710 0,4
%A A120710 Marc LeBrun (mlb(AT)well.com), Jun 28 2006

%I A115780
%S A115780 2,0,4,8,14,32,60,140,212,750,1322,2540,6862,13040,27174,57052,117164,
%T A115780 248360,555254
%N A115780 Consider the Levenshtein distance between k considered as a decimal string and k considered as a binary string. Then a(n) is the number of nonnegative integers having a Levenshtein distance of n.
%C A115780 a(n)~2^n. a(n)-2^n: -1,2,0,0,2,0,4,12,44,238,298,492,2766,4848,10790,24284,51628,117288,293110, ...,.
%e A115780 a(0)=2 since only 0&1 have a Levenshtein distance of zero when considering them as decimal and binary strings,
%t A115780 levenshtein[s_List, t_List] := Module[{d, n = Length@s, m = Length@t}, Which[s === t, 0, n == 0, m, m == 0, n, s != t, d = Table[0, {m + 1}, {n + 1}]; d[[1, Range[n + 1]]] = Range[0, n]; d[[Range[m + 1], 1]] = Range[0, m]; Do[d[[j + 1, i + 1]] = Min[d[[j, i + 1]] + 1, d[[j + 1, i]] + 1, d[[j, i]] + If[s[[i]] === t[[j]], 0, 1]], {j, m}, {i, n}]; d[[ -1, -1]]]];
%t A115780 t = Table[0, {25}]; f[n_] := levenshtein[ IntegerDigits[n], IntegerDigits[n, 2]]; Do[ t[[f@n+1]]++, {n, 10^6}]; t
%Y A115780 Cf. A000027, A007088, A115777.
%Y A115780 Adjacent sequences: A115777 A115778 A115779 this_sequence A115781 A115782 A115783
%Y A115780 Sequence in context: A120558 A120554 A120710 this_sequence A101189 A070015 A021492
%K A115780 more,nonn
%O A115780 0,1
%A A115780 Robert G. Wilson v (rgwv(AT)rgwv.com), Jan 26 2006

%I A101189
%S A101189 1,2,0,4,8,16,40,144,512,1696,5696,19840,70048,247744,880128,3152768,11386624,
%T A101189 41389568,151273728,555794944,2052141056,7610274816,28331018240,105833345024,
%U A101189 396594444800,1490425179136,5615651143680,21209004267520,80276663808000
%V A101189 1,2,0,4,-8,16,-40,144,-512,1696,-5696,19840,-70048,247744,-880128,3152768,-11386624,
%W A101189 41389568,-151273728,555794944,-2052141056,7610274816,-28331018240,105833345024,
%X A101189 -396594444800,1490425179136,-5615651143680,21209004267520,-80276663808000
%N A101189 G.f. defined as the limit: A(x) = limit_{n->oo} F(n)^(1/2^(n-1)) where F(n) is the n-th iteration of: F(0) = 1, F(n) = F(n-1)^2 + (2x)^(2^n-1) for n>=1.
%C A101189 The coefficients of x^n in A(x/2)^(1/2) equals A101190(n)/2^A005187(n). The coefficients of x^n in A(x/2)^(1/4) equals A101191(n)/2^A004134(n). A101190 and A101191 are related to doubly exponential numbers A003095 and to Catalan numbers (A000108).
%F A101189 G.f. A(x) = [Sum_{n>=0} A101190(n)/2^A005187(n)*(2x)^n]^2. G.f. A(x) = [Sum_{n>=0} A101191(n)/2^A004134(n)*(2x)^n]^4.
%e A101189 The iteration begins:
%e A101189 F(0) = 1,
%e A101189 F(1) = F(0)^2 + (2*x)^(2^1-1)
%e A101189 = 1 +2*x,
%e A101189 F(2) = F(1)^2 + (2*x)^(2^2-1)
%e A101189 = 1 +4*x +4*x^2 +8*x^3,
%e A101189 F(3) = F(2)^2 + (2*x)^(2^3-1)
%e A101189 = 1 +8*x +24*x^2 +48*x^3 +80*x^4 +64*x^5 +64*x^6 +128*x^7.
%e A101189 The 2^(n-1)-th roots of F(n) tend to the limit of A(x):
%e A101189 F(1)^(1/2^0) = 1 +2*x
%e A101189 F(2)^(1/2^1) = 1 +2*x +4*x^3 -8*x^4 +16*x^5 -40*x^6 + ...
%e A101189 F(3)^(1/2^2) = 1 +2*x +4*x^3 -8*x^4 +16*x^5 -40*x^6 +144*x^7 -512*x^8 +...
%e A101189 The limit of this process is the g.f. A(x).
%o A101189 (PARI) {a(n)=local(F=1,A,L);if(n==0,A=1,L=ceil(log(n+1)/log(2)); for(k=1,L,F=F^2+(2*x)^(2^k-1));A=polcoeff(F^(1/(2^(L-1)))+x*O(x^n),n));A}
%Y A101189 Cf. A101190, A101191, A005187, A004134, A003095.
%Y A101189 Adjacent sequences: A101186 A101187 A101188 this_sequence A101190 A101191 A101192
%Y A101189 Sequence in context: A120554 A120710 A115780 this_sequence A070015 A021492 A077119
%K A101189 sign
%O A101189 0,2
%A A101189 Paul D. Hanna (pauldhanna(AT)juno.com), Dec 03 2004

%I A070015
%S A070015 2,0,4,9,0,6,8,10,15,14,21,121,27,22,16,12,39,289,65,34,18,20,57,529,
%T A070015 95,46,69,28,115,841,32,58,45,62,93,24,155,1369,217,44,63,30,50,82,123,
%U A070015 52,129,2209,75,40,141,0,235,42,36,106,99,68,265,3481,371,118,64,56
%N A070015 Least m such that sum of aliquot parts of m [A001065(m)] equals n or 0 if no such number exists.
%H A070015 Richard J Mathar, <a href="http://www.research.att.com/~njas/sequences/b070015.txt">Table of n, a(n) for n = 1..9884</a>
%F A070015 a(n)=Min{x; A001065(x)=n} or a(n)=0 if n is untouchable number (i.e. if from A005114)
%e A070015 n=128: a(n)=16129, divisors={1,127,16129}, 1+127=sigma[n]-n=128 and 16129 is the smallest.
%t A070015 f[x_] := DivisorSigma[1, x]-x; t=Table[0, {128}]; Do[c=f[n]; If[c<129&&t[[c]]==0, t[[c]]=n], {n, 1, 1000000}]; t
%Y A070015 Cf. A000203, A001065, A048050, A051444, A007369, A070016, A005114, A048995.
%Y A070015 Adjacent sequences: A070012 A070013 A070014 this_sequence A070016 A070017 A070018
%Y A070015 Sequence in context: A120710 A115780 A101189 this_sequence A021492 A077119 A002938
%K A070015 nonn
%O A070015 1,1
%A A070015 Labos E. (labos(AT)ana.sote.hu), Apr 12 2002

%I A021492
%S A021492 0,0,2,0,4,9,1,8,0,3,2,7,8,6,8,8,5,2,4,5,9,0,1,6,3,9,3,4,4,2,6,2,2,
%T A021492 9,5,0,8,1,9,6,7,2,1,3,1,1,4,7,5,4,0,9,8,3,6,0,6,5,5,7,3,7,7,0,4,9,
%U A021492 1,8,0,3,2,7,8,6,8,8,5,2,4,5,9,0,1,6,3,9,3,4,4,2,6,2,2,9,5,0,8,1,9
%N A021492 Decimal expansion of 1/488.
%Y A021492 Adjacent sequences: A021489 A021490 A021491 this_sequence A021493 A021494 A021495
%Y A021492 Sequence in context: A115780 A101189 A070015 this_sequence A077119 A002938 A111938
%K A021492 nonn,cons
%O A021492 0,3
%A A021492 njas

%I A077119
%S A077119 0,0,1,2,0,4,9,18,17,0,24,35,36,12,40,11,0,13,56,30,79,45,39,67,100,0,
%T A077119 113,83,48,53,104,138,7,163,100,26,0,28,116,217,9,248,104,17,80,79,8,
%U A077119 139,297,0,316,155,17,119,145,89,55
%V A077119 0,0,1,-2,0,-4,9,18,17,0,24,-35,36,12,-40,-11,0,-13,-56,30,-79,-45,-39,-67,100,0,113,
%W A077119 -83,-48,-53,-104,138,-7,163,-100,-26,0,-28,-116,217,9,248,-104,17,80,79,8,-139,297,0,
%X A077119 316,-155,17,119,145,89,-55
%N A077119 A077118(n) - n^3.
%C A077119 a(n)=0 iff n = m^(6*k).
%F A077119 a(n) = if A077116(n)<A070929(n) then -A077116(n) else A070929(n).
%e A077119 A077118(10)=1024=32^2 is the nearest square to 10^3=1000, therefore a(10)=1024-1000=24.
%Y A077119 Cf. A000578, A077118, A077111.
%Y A077119 |a(n)| = A002938(n).
%Y A077119 Adjacent sequences: A077116 A077117 A077118 this_sequence A077120 A077121 A077122
%Y A077119 Sequence in context: A101189 A070015 A021492 this_sequence A002938 A111938 A055978
%K A077119 sign
%O A077119 0,4
%A A077119 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Oct 29 2002

%I A002938 M0033 N0008
%S A002938 0,1,2,0,4,9,18,17,0,24,35,36,12,40,11,0,13,56,30,79,45,39,67,100,0,
%T A002938 113,83,48,53,104,138,7,163,100,26,0,28,116,217,9,248,104,17,80,79,8,
%U A002938 139,297,0,316,155,17,119,145,89,55,293,252,170,225,405,184,47,0,49
%N A002938 The minimal sequence (from solving n^3 - m^2 = a(n)).
%C A002938 a(n^2)=0 - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 17 2002
%D A002938 Marshall Hall, Jr., The diophantine equation x^3-y^2=k, pp. 173-198 of A. O. L. Atkin and B. J. Birch, editors, Computers in Number Theory. Academic Press, NY, 1971.
%o A002938 (PARI) a(n)=vecmin(vector(ceil(n^(3/2)),i,abs(n^3-i^2)))
%Y A002938 a(n) = |A077119(n)|.
%Y A002938 Adjacent sequences: A002935 A002936 A002937 this_sequence A002939 A002940 A002941
%Y A002938 Sequence in context: A070015 A021492 A077119 this_sequence A111938 A055978 A069025
%K A002938 nonn
%O A002938 1,3
%A A002938 njas
%E A002938 More terms from Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 17 2002

%I A111938
%S A111938 1,2,0,4,10,0,0,8,9,20,0,0,26,0,0,16,34,18,0,40,0,0,0,0,75,52,0,0,58,0,
%T A111938 0,32,0,68,0,36,74,0,0,80,82,0,0,0,90,0,0,0,49,150,0,104,106,0,0,0,0,
%U A111938 116,0,0,122,0,0,64,260,0,0,136,0,0,0,72,146,148,0,0,0,0,0,160,81,164
%N A111938 n times number of divisors of n of form 4m+1 - n times number of divisors of form 4m+3.
%F A111938 Multiplicative with a(p^e) = p^e if p = 2; (e+1)p^e if p == 1 (mod 4); (1+(-1)^e)/2 p^e if p == 3 (mod 4).
%F A111938 G.f.: Sum_{k>0} k(x^k-x^(3k))/(1+x^(2k))^2 = Sum_{k>0} -(-1)^k(2k-1)x^(2k-1)/(1-x^(2k-1))^2.
%F A111938 G.f.: xd/dx(theta_3(x)^2)/4 . - Michael Somos Nov 07 2005
%F A111938 G.f.: (1/4)* Sum_{u,v} (u*u +v*v)* x^(u*u +v*v). - Michael Somos Jun 14 2007
%o A111938 (PARI) a(n)=if(n<1, 0, n*sumdiv(n,d, (d%4==1)-(d%4==3)))
%o A111938 (PARI) {a(n)=local(r); if(n<1, 0, r=sqrtint(n); sum(x=-r,r, sum(y=-r,r, if(x^2+y^2==n, (x+y)^2) ))/4 )} /* Michael Somos Sep 12 2005 */
%o A111938 (PARI) {a(n)=if(n<1, 0, n*polcoeff( sum(k=1,sqrtint(n), 2*x^k^2, 1+x*O(x^n))^2, n)/4 )} /* Michael Somos Sep 12 2005 */
%Y A111938 n*A002654(n)=a(n).
%Y A111938 Adjacent sequences: A111935 A111936 A111937 this_sequence A111939 A111940 A111941
%Y A111938 Sequence in context: A021492 A077119 A002938 this_sequence A055978 A069025 A066442
%K A111938 nonn,mult
%O A111938 1,2
%A A111938 Michael Somos, Aug 21 2005

%I A055978
%S A055978 1,2,0,4,24,36,0,64,252,290,0,396,1472,1380,0,944,4830,4248,0,1268,6048,8040,
%T A055978 0,12528,16744,3706,0,20976,84480,31284,0,31312,113643,101542,0,152892,115920,
%U A055978 104792,0,96576,534612,112914,0,369544,370944,334864,0,603936,577738,22554,0
%V A055978 1,-2,0,4,-24,36,0,-64,252,-290,0,396,-1472,1380,0,-944,4830,-4248,0,-1268,-6048,8040,
%W A055978 0,12528,-16744,-3706,0,-20976,84480,-31284,0,-31312,-113643,101542,0,152892,-115920,
%X A055978 -104792,0,-96576,534612,-112914,0,-369544,-370944,334864,0,603936,-577738,-22554,0
%N A055978 A sequence related to Ramanujan's tau function.
%D A055978 Eichler and Zagier, The Theory of Jacobi Forms, Birkhauser, 1985.
%F A055978 a(4n+2)=0, a(4n)=A000594(n) (Ramanujan tau(n)).
%F A055978 Sum_{k>0} a(4k+1)q^(4k+1) = (-1)(q*d/dq theta_2(q^4))*eta(q^4)^18*eta(q^16)^2/eta(q^8). - Michael Somos Mar 20 2004
%F A055978 Sum_{k>0} a(4k+3)q^(4k+3) = (1/2)(q*d/dq theta_3(q^4))*eta(q^4)^16*eta(q^8)^5/eta(q^16)^2. - Michael Somos Mar 20 2004
%F A055978 G.f.: x^3(Product_{k>0} (1-x^k)(1-x^(4k))^18/(1+x^k))(Sum_{k>0} k^2 x^(k^2)). - Michael Somos Mar 20 2004
%F A055978 phi_{10, 1}*q*(d/dq){theta_3(z)} where phi_{10, 1} is unique Jacobi cusp form of weight 10 index 1 given by A003784.
%o A055978 (PARI) a(n)=if(n<3,0,n-=3; X=x+x*O(x^n); polcoeff(eta(X)^2*eta(X^4)^18/eta(X^2)*sum(k=1,sqrtint(n),k^2*x^(k^2)),n))
%Y A055978 A003784, A000594.
%Y A055978 Adjacent sequences: A055975 A055976 A055977 this_sequence A055979 A055980 A055981
%Y A055978 Sequence in context: A077119 A002938 A111938 this_sequence A069025 A066442 A086134
%K A055978 sign
%O A055978 4,2
%A A055978 Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jul 24 2000

%I A069025
%S A069025 1,2,0,4,32,0,16,8,0,64,128,0,256,2048,0,0,0,0,4096,8192,0,16384,0,0,
%T A069025 65536,32768,0,0,524288,0,1048576,0,0,0,134217728,0,16777216,0,0,
%U A069025 67108864,8388608,0,268435456,0,0,4398046511104,2147483648,0,0
%N A069025 Smallest power of 2 with digital sum (A007953) n, or 0 if no such number exists.
%C A069025 a(3k)=0. In general about half the entries are nonzero.
%e A069025 Both 2^4=16 and 2^10=1024 have a digital sum of 7 but 2^4 is the smaller so it is the one presented.
%t A069025 a = Table[0, {50}]; Do[b = Plus @@ IntegerDigits[2^n]; If[b < 51 && a[[b]] == 0, a[[b]] = 2^n], {n, 0, 10^4}]; a
%Y A069025 Cf. A007632.
%Y A069025 Adjacent sequences: A069022 A069023 A069024 this_sequence A069026 A069027 A069028
%Y A069025 Sequence in context: A002938 A111938 A055978 this_sequence A066442 A086134 A071090
%K A069025 base,nonn
%O A069025 1,2
%A A069025 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 02 2002
%E A069025 Edited by Robert G. Wilson v (rgwv(AT)rgwv.com), Nov 05 2002

%I A066442
%S A066442 0,0,0,0,2,0,5,0,0,4,1,0,12,4,3,0,12,0,12,16,6,12,12,0,7,14,0,16,12,24,
%T A066442 12,0,12,8,3,0,12,30,12,16,12,36,12,12,27,6,12,0,19,24,45,40,12,0,23,
%U A066442 32,18,28,12,36,12,20,27,0,12,12,12,64,3,44,12,0,12,70,18,64,45,66,12
%N A066442 12^n mod n.
%t A066442 Table[PowerMod[12, n, n], {n, 80} ]
%Y A066442 Adjacent sequences: A066439 A066440 A066441 this_sequence A066443 A066444 A066445
%Y A066442 Sequence in context: A111938 A055978 A069025 this_sequence A086134 A071090 A105221
%K A066442 nonn
%O A066442 1,5
%A A066442 Robert G. Wilson v (rgwv(AT)rgwv.com), Dec 27 2001

%I A086134
%S A086134 0,0,0,2,0,5,0,2,2,7,0,2,0,3,2,2,0,3,0,2,2,13,0,2,2,3,3,2,0,31,0,2,2,19,
%T A086134 2,2,0,3,2,2,0,41,0,2,3,5,0,2,2,3,2,2,0,3,2,2,2,31,0,2,0,3,3,2,2,61,0,2,
%U A086134 2,59,0,2,0,3,5,2,2,71,0,2,2,43,0,2,2,3,2,2,0,3,2,2,2,7,2,2,0,7,3,2,0,7
%N A086134 Smallest prime factor of arithmetic derivative of n or a(n)=0 if no such prime exists.
%Y A086134 Cf. A003415.
%Y A086134 Adjacent sequences: A086131 A086132 A086133 this_sequence A086135 A086136 A086137
%Y A086134 Sequence in context: A055978 A069025 A066442 this_sequence A071090 A105221 A061376
%K A086134 nonn
%O A086134 1,4
%A A086134 Labos E. (labos(AT)ana.sote.hu), Jul 23 2003

%I A071090
%S A071090 1,1,0,2,0,5,0,2,3,0,0,7,0,0,8,4,0,3,0,9,0,0,0,10,5,0,0,11,0,11,0,4,0,
%T A071090 0,12,6,0,0,0,13,0,13,0,0,14,0,0,14,7,5,0,0,0,15,0,15,0,0,0,16,0,0,16,
%U A071090 8,0,17,0,0,0,17,0,23,0,0,0,0,18,0,0,18,9,0,0,19,0,0,0,19,0,19,20,0,0
%N A071090 Sum of middle divisors of n.
%C A071090 Divisors are in the half-open interval [sqrt(n/2), sqrt(n*2)).
%t A071090 Table[Plus @@ Select[ Divisors[n], Sqrt[n/2] <= # < Sqrt[n*2] &], {n, 1, 95}]
%Y A071090 Cf. A067742.
%Y A071090 Adjacent sequences: A071087 A071088 A071089 this_sequence A071091 A071092 A071093
%Y A071090 Sequence in context: A069025 A066442 A086134 this_sequence A105221 A061376 A058974
%K A071090 nonn,easy
%O A071090 1,4
%A A071090 njas, May 27 2002
%E A071090 Extended by Robert G. Wilson v (rgwv(AT)rgwv.com), May 30 2002

%I A105221
%S A105221 0,0,0,2,0,5,0,2,3,7,0,5,0,9,8,2,0,5,0,7,10,13,0,5,5,15,3,9,0,10,0,2,14,
%T A105221 19,12,5,0,21,16,7,0,12,0,13,8,25,0,5,7,7,20,15,0,5,16,9,22,31,0,10,0,
%U A105221 33,10,2,18,16,0,19,26,14,0,5,0,39,8,21,18,18,0,7,3,43,0,12,22,45
%N A105221 a(n) = the sum of n's distinct prime factors below n.
%H A105221 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b105221.txt">Table of n, a(n) for n=1..1000</a>
%e A105221 a(12)=5 because 12's distinct prime factors 2 and 3 sum to 5.
%Y A105221 Cf. A003508.
%Y A105221 Cf. A008472
%Y A105221 Adjacent sequences: A105218 A105219 A105220 this_sequence A105222 A105223 A105224
%Y A105221 Sequence in context: A066442 A086134 A071090 this_sequence A061376 A058974 A019962
%K A105221 easy,nonn
%O A105221 1,4
%A A105221 Alexandre Wajnberg (alexandre.wajnberg(AT)ulb.ac.be), Apr 13 2005
%E A105221 Edited by Don Reble (djr(AT)nk.ca), Nov 17 2005

%I A061376
%S A061376 0,0,0,2,0,5,0,2,3,7,0,5,0,12,10,2,0,5,0,7,17,13,0,5,5,23,3,12,0,17,0,
%T A061376 2,23,19,17,5,0,31,18,7,0,17,0,13,10,30,0,5,7,7,27,23,0,5,18,12,35,31,
%U A061376 0,17,0,47,17,2,23,18,0,19,41,23,0,5,0,55,10,31,23,23,0,7
%N A061376 a(n) = f(n) + f(f(n)) where f(n) = 0 if n = 1 or a prime, otherwise f(n) = sum of distinct primes of n.
%C A061376 Note that this sequence differs from A058974 at n = 26, 33, 38, 52, 62, 69, 70, 74, 76, 86, 99, etc.
%e A061376 a(14) = 12 because f(14) = 2+7 = 9 and f(9) = 3 and 9+3 = 12.
%t A061376 f[n_Integer] := If[n == 1 || PrimeQ[n], 0, Plus@@First[ Transpose[ FactorInteger[n] ] ] ]; Table[ f[n] + f[f[n]], {n, 1, 80} ]
%Y A061376 Cf. A008472, A058974.
%Y A061376 Adjacent sequences: A061373 A061374 A061375 this_sequence A061377 A061378 A061379
%Y A061376 Sequence in context: A086134 A071090 A105221 this_sequence A058974 A019962 A086131
%K A061376 nonn
%O A061376 1,4
%A A061376 Robert G. Wilson v (rgwv(AT)rgwv.com), Jun 08 2001

%I A058974
%S A058974 0,0,0,2,0,5,0,2,3,7,0,5,0,12,10,2,0,5,0,7,17,13,0,5,5,25,3,12,
%T A058974 0,17,0,2,26,19,17,5,0,38,18,7,0,17,0,13,10,30,0,5,7,7,27,25,0,
%U A058974 5,18,12,35,31,0,17,0,59,17,2,23,18,0,19,51,26,0,5,0,57,10,38,23
%N A058974 a(n) = 0 if n = 1 or a prime, otherwise a(n) = s + a(s) iterated until no change occurs, where s (A008472) is sum of distinct primes dividing n.
%D A058974 E. N. Gilbert, An interesting property of 38, unpublished, circa 1992. Shows that 38 is the only solution of a(n) = n.
%p A058974 f := proc(n) option remember; local i,j,k,t1,t2; if n = 1 or isprime(n) then 0 else A008472(n) + f(A008472(n)); fi; end;
%t A058974 f[n_Integer] := If[n == 1 || PrimeQ[n], 0, Plus @@ First[ Transpose[ FactorInteger[n]]]]; Table[Plus @@ Drop[ FixedPointList[f, n], 1], {n, 1, 80}]
%Y A058974 Cf. A008472.
%Y A058974 Adjacent sequences: A058971 A058972 A058973 this_sequence A058975 A058976 A058977
%Y A058974 Sequence in context: A071090 A105221 A061376 this_sequence A019962 A086131 A104755
%K A058974 nonn
%O A058974 1,4
%A A058974 njas, Jan 15 2001

%I A019962
%S A019962 2,0,5,0,3,0,3,8,4,1,5,7,9,2,9,6,2,1,6,8,9,9,0,1,1,0,7,0,5,4,1,4,9,
%T A019962 4,1,4,6,7,6,7,5,1,9,6,2,2,7,4,3,2,4,2,4,2,3,4,7,2,6,6,6,0,9,6,7,8,
%U A019962 5,4,8,1,1,4,4,7,7,0,6,5,7,7,4,2,9,4,9,7,7,0,8,8,6,9,4,2,9,1,6,8,1
%N A019962 Decimal expansion of tangent of 64 degrees.
%Y A019962 Adjacent sequences: A019959 A019960 A019961 this_sequence A019963 A019964 A019965
%Y A019962 Sequence in context: A105221 A061376 A058974 this_sequence A086131 A104755 A054013
%K A019962 nonn,cons
%O A019962 1,1
%A A019962 njas

%I A086131
%S A086131 0,0,0,2,0,5,0,3,3,7,0,2,0,3,2,2,0,7,0,3,5,13,0,11,5,5,3,2,0,31,0,5,7,
%T A086131 19,3,5,0,7,2,17,0,41,0,3,13,5,0,7,7,5,5,7,0,3,2,23,11,31,0,23,0,11,17,
%U A086131 3,3,61,0,3,13,59,0,13,0,13,11,5,3,71,0,11,3,43,0,31,11,5,2,7,0,41,5,3
%N A086131 Largest prime factor of arithmetic derivative of n if it exists, or a(n)=0 for n=1 and n=prime.
%Y A086131 Cf. A003415.
%Y A086131 Adjacent sequences: A086128 A086129 A086130 this_sequence A086132 A086133 A086134
%Y A086131 Sequence in context: A061376 A058974 A019962 this_sequence A104755 A054013 A048050
%K A086131 nonn
%O A086131 1,4
%A A086131 Labos E. (labos(AT)ana.sote.hu), Jul 23 2003

%I A104755
%S A104755 1,2,0,5,0,5,3,4,2,6,1,9,6,3,9,1,7,4,9,3,3,8,0,9,6,3,6,5,6,5,4,9,3,2,1,
%T A104755 0,6,8,6,6,5,1
%N A104755 Decimal expansion of solution to x^(7^x)=7.
%F A104755 x=1.2050534261963917; x^(7^x)=7
%Y A104755 Cf. A103561, A104750-A104761.
%Y A104755 Adjacent sequences: A104752 A104753 A104754 this_sequence A104756 A104757 A104758
%Y A104755 Sequence in context: A058974 A019962 A086131 this_sequence A054013 A048050 A078153
%K A104755 cons,nonn
%O A104755 1,2
%A A104755 Zak Seidov (zakseidov(AT)yahoo.com), Mar 23 2005

%I A054013
%S A054013 0,0,0,2,0,5,0,6,3,7,0,3,0,9,8,14,0,2,0,1,10,13,0,11,5,15,12,27,0,11,0,
%T A054013 30,14,19,12,18,0,21,16,9,0,11,0,39,32,25,0,27,7,42,20,45,0,11,16,7,22,
%U A054013 31,0,47,0,33,40,62,18,11,0,57,26,3,0,50,0,39,48,63,18,11,0,25,39,43,0
%N A054013 Chowla function of n read modulo n.
%C A054013 Chowla's function (A048050) = sum of divisors of n except 1 and n.
%F A054013 a(n) = A048050(n) mod n
%p A054013 with(numtheory): [seq((sigma(i) - i - 1) mod i, i=2..100)];
%Y A054013 Cf. A048050, A054014, A054015.
%Y A054013 Adjacent sequences: A054010 A054011 A054012 this_sequence A054014 A054015 A054016
%Y A054013 Sequence in context: A019962 A086131 A104755 this_sequence A048050 A078153 A104035
%K A054013 nonn
%O A054013 1,4
%A A054013 Asher Auel (asher.auel(AT)reed.edu) Jan 17, 2000

%I A048050
%S A048050 0,0,0,2,0,5,0,6,3,7,0,15,0,9,8,14,0,20,0,21,10,13,0,35,5,15,12,27,
%T A048050 0,41,0,30,14,19,12,54,0,21,16,49,0,53,0,39,32,25,0,75,7,42,20,
%U A048050 45,0,65,16,63,22,31,0,107,0,33,40,62,18,77,0,57,26,73,0,122,0
%N A048050 Chowla's function: sum of divisors of n except 1 and n.
%D A048050 M. Lal and A. Forbes, A note on Chowla's function, Math. Comp., 25 (1971), 923-925.
%H A048050 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b048050.txt">Table of n, a(n) for n=1..10000</a>
%e A048050 Divisors of 20 are 1,2,4,5,10,20, so a(20)=2+4+5+10=21.
%p A048050 with(numtheory); n->sigma(n)-n-1; # n>1
%Y A048050 Cf. A001065, A000593, A002954, A048995.
%Y A048050 Adjacent sequences: A048047 A048048 A048049 this_sequence A048051 A048052 A048053
%Y A048050 Sequence in context: A086131 A104755 A054013 this_sequence A078153 A104035 A115333
%K A048050 nonn,nice,easy
%O A048050 1,4
%A A048050 njas

%I A078153
%S A078153 0,0,0,0,2,0,5,0,6,3,10,0,15,7,9,8,22,4,24,9,21,19,32,0,35,26,30,17,44,
%T A078153 11,52,24,41,37,45,12,66,46,52,22,71,27,80,43,52,60,85,14,89,56,79,56,
%U A078153 101,39,89,52,94,86,117,15,122,90,85,73,118,62,139,84,116,72,145,36
%N A078153 a(n)=A051201[n]-A000203[n].
%e A078153 n=15: sequence of D1={Floor[15/j]}={15,7,5,3,3,2,2,1,1,1,1,1,1,1,1}, Union[D1]={15,7,5,3,2,1}=Divisors[15]and{7,2}, a[15]=(15+7+5+3+2+1)-sigma[15]=7+2=9.
%t A078153 Table[Apply[Plus, Union[Table[Floor[w/j], {j, 1, w}]]] -DivisorSigma[1, w], {w, 1, 128}]
%Y A078153 Cf. A051201, A000203, A055086, A000005, A078152, A076891.
%Y A078153 Adjacent sequences: A078150 A078151 A078152 this_sequence A078154 A078155 A078156
%Y A078153 Sequence in context: A104755 A054013 A048050 this_sequence A104035 A115333 A105523
%K A078153 nonn
%O A078153 1,5
%A A078153 Labos E. (labos(AT)ana.sote.hu), Nov 27 2002

%I A104035
%S A104035 1,0,1,1,0,2,0,5,0,6,5,0,28,0,24,0,61,0,180,0,120,61,0,662,0,1320,0,720,
%T A104035 0,1385,0,7266,0,10920,0,5040,1385,0,24568,0,83664,0,100800,0,40320,0,
%U A104035 50521,0,408360,0,1023120,0,1028160,0,362880,50521,0,1326122,0,6749040
%N A104035 Triangle T(n,k), 0<=k<=n, read by rows, defined by T(0,0) = 1; T(0,k) = 0 if k>0 or if k<0; T(n,k) = k*T(n-1,k-1) + (k+1)*T(n-1,k+1).
%C A104035 Triangle related to Euler and Springer numbers.
%D A104035 S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see pp. 445 and 469.
%F A104035 T(n, n) = n!; T(n, 0) = 0 if n = 2m + 1; T(n, 0) = A000364(m) if n = 2m.
%F A104035 Sum_{k>=0} T(m, k)*T(n, k) = T(m+n, 0).
%F A104035 Sum_{k>=0} T(n, k) = A001586(n): Springer numbers.
%e A104035 Triangle begins:
%e A104035 1
%e A104035 0 1
%e A104035 1 0 2
%e A104035 0 5 0 6
%e A104035 5 0 28 0 24
%e A104035 0 61 0 180 0 120
%e A104035 61 0 662 0 1320 0 720
%e A104035 0 1385 0 7266 0 10920 0 5040
%Y A104035 Cf. A000364 A001586.
%Y A104035 Adjacent sequences: A104032 A104033 A104034 this_sequence A104036 A104037 A104038
%Y A104035 Sequence in context: A054013 A048050 A078153 this_sequence A115333 A105523 A126120
%K A104035 nonn,easy,tabl
%O A104035 0,6
%A A104035 Philippe DELEHAM ( kolotoko(AT)wanadoo.fr), Apr 06 2005

%I A115333
%S A115333 0,0,2,0,5,0,10,0,2,3,17,0,28,8,2,0,41,0,58,3,7,15,77,0,5,26,2,8,100,0,
%T A115333 129,0,14,39,5,0,160,56,25,3,197,5,238,15,2,75,281,0,10,3,38,26,328,0,
%U A115333 12,8,55,98,381,0,440,127,7,0,23,12,501,39,74,3,568,0,639,158,2,56,10
%N A115333 Sum of primes which do not divide n and are less than the largest prime dividing n.
%C A115333 When n is prime, n = largest prime dividing n; hence a(n) is the sum of all primes less than n = A034387(n)-n. a(n) = SUM{p such that p is in A000040 AND NOT(p|n) AND p<A006530(n)}. - Jonathan Vos Post (jvospost2(AT)yahoo.com), Mar 08 2006
%C A115333 The zeros give A055932: All prime divisors are consecutive primes starting at 2. - Robert G. Wilson v May 01 2006
%e A115333 The primes < 7 and coprime to 7 are 2, 3, and 5. So a(7) = 2+3+5 = 10.
%t A115333 f[n_] := Plus @@ Complement[Prime@ Range@ PrimePi[ Max[First /@ FactorInteger@n] - 1], First /@ FactorInteger@n]; Array[f, 77] - Hans Havermann (pxp(AT)rogers.com), Mar 06 2006
%Y A115333 Cf. A000040, A006530, A034387, A066911, A083720.
%Y A115333 Adjacent sequences: A115330 A115331 A115332 this_sequence A115334 A115335 A115336
%Y A115333 Sequence in context: A048050 A078153 A104035 this_sequence A105523 A126120 A090192
%K A115333 nonn
%O A115333 1,3
%A A115333 Leroy Quet (qq-quet(AT)mindspring.com), Mar 05 2006
%E A115333 More terms from Hans Havermann (pxp(AT)rogers.com), Mar 06 2006

%I A105523
%S A105523 1,1,0,1,0,2,0,5,0,14,0,42,0,132,0,429,0,1430,0,4862,0,16796,0,58786,0,
%T A105523 208012,0,742900,0,2674440,0,9694845,0,35357670,0,129644790,0,477638700,
%U A105523 0,1767263190,0
%V A105523 1,-1,0,1,0,-2,0,5,0,-14,0,42,0,-132,0,429,0,-1430,0,4862,0,-16796,0,58786,0,-208012,0,
%W A105523 742900,0,-2674440,0,9694845,0,-35357670,0,129644790,0,-477638700,0,1767263190,0
%N A105523 Expansion of 1-xc(-x^2) where c(x) is the g.f. of A000108.
%C A105523 Row sums of A105522. Row sums of inverse of A105438.
%C A105523 First column of number triangle A106180.
%F A105523 G.f.: (1+2x-sqrt(1+4x^2))/(2x)
%F A105523 a(n)=0^n+sin(pi*(n-2)/2)(C((n-1)/2)(1-(-1)^n)/2);
%Y A105523 Cf. A097331, A090192.
%Y A105523 Adjacent sequences: A105520 A105521 A105522 this_sequence A105524 A105525 A105526
%Y A105523 Sequence in context: A078153 A104035 A115333 this_sequence A126120 A090192 A097331
%K A105523 easy,sign
%O A105523 0,6
%A A105523 Paul Barry (pbarry(AT)wit.ie), Apr 11 2005

%I A126120
%S A126120 1,0,1,0,2,0,5,0,14,0,42,0,132,0,429,0,1430,0,4862,0,16796,0,58786,0,
%T A126120 208012,0,742900,0,2674440,0,9694845,0,35357670,0,129644790,0,477638700,
%U A126120 0,1767263190,0,6564120420,0,24466267020,0,91482563640,0
%N A126120 A000108 (Catalan numbers) interpolated with 0's.
%C A126120 Inverse binomial transform of A001006.
%C A126120 The Hankel transform of this sequence gives A000012 = [1,1,1,1,1,...].
%C A126120 Counts returning walks of length n on a 1-d integer lattice with step set {+1,-1} which stay in the chamber x >= 0. - Andrew V. Sutherland (drew(AT)math.mit.edu), Feb 29 2008
%C A126120 Moment sequence of the trace of a random matrix in G=USp(2)=SU(2). If X=tr(A) is a random variable (A distributed according to the Haar measure on G) then a(n) = E[X^n]. - Andrew V. Sutherland (drew(AT)math.mit.edu), Feb 29 2008
%C A126120 a(n) is the coefficient of z^n in I_0(2z), where I_0 is the hyperbolic Bessel function (of the first kind) of order zero. - Andrew V. Sutherland (drew(AT)math.mit.edu), Mar 16 2008
%D A126120 Martin Aigner, "Catalan and other numbers: a recurrent theme", in Algebraic Combinatorics and Computer Science, a Tribute to Gian-Carlo Rota, pp.347-390, Springer, 2001.
%D A126120 Kiran S. Kedlaya and Andrew V. Sutherland, "Hyperelliptic curves, L-polynomials, and random matrices", preprint, 2008.
%D A126120 Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Ch. 49, Hemisphere Publishing Corp., 1987.
%H A126120 Kiran S. Kedlaya and Andrew V. Sutherland, <a href="http://arxiv.org/abs/0803.4462">Hyperelliptic curves, L-polynomials, and random matrices</a>.
%F A126120 a(2*n)=A000108(n), a(2*n+1)=0 . a(n)=A053121(n,0).
%F A126120 (1/Pi) Integral_{0 .. Pi } (2cos(x))^n*2sin^2(x) dx. - Andrew V. Sutherland (drew(AT)math.mit.edu), Feb 29 2008
%p A126120 with(combstruct):grammar := { BB = Sequence(Prod(a,BB,b)), a = Atom, b = Atom }: > seq(count([BB,grammar],size=n),n=0..47); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 25 2007
%p A126120 BB:={E=Prod(Z,Z),S=Union(Epsilon,Prod(S,S,E))}: ZL:=[S,BB,unlabeled]: > seq(count(ZL,size=n),n=0..45); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2007
%p A126120 BB:=[T,{T=Prod(Z,Z,Z,F,F),F=Sequence(B),B=Prod(F,Z,Z)}, unlabeled]: seq(count(BB,size=i),i=1..46); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2007
%Y A126120 Cf. A000108.
%Y A126120 Adjacent sequences: A126117 A126118 A126119 this_sequence A126121 A126122 A126123
%Y A126120 Sequence in context: A104035 A115333 A105523 this_sequence A090192 A097331 A094032
%K A126120 nonn
%O A126120 0,5
%A A126120 Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 06 2007

%I A090192
%S A090192 1,1,0,1,0,2,0,5,0,14,0,42,0,132,0,429,0,1430,0,4862,0,16796,0,58786,0,
%T A090192 208012,0,742900,0,2674440,0,9694845,0,35357670,0,129644790,0,477638700,
%U A090192 0,1767263190,0,6564120420,0,24466267020,0,91482563640,0,343059613650,0
%V A090192 1,1,0,-1,0,2,0,-5,0,14,0,-42,0,132,0,-429,0,1430,0,-4862,0,16796,0,-58786,0,208012,0,
%W A090192 -742900,0,2674440,0,-9694845,0,35357670,0,-129644790,0,477638700,0,-1767263190,0,
%X A090192 6564120420,0,-24466267020,0,91482563640,0,-343059613650,0
%N A090192 q-Catalan numbers (recurrence version) for q= -1.
%C A090192 Hankel transform is (-1)^C(n+1,2). - Paul Barry (pbarry(AT)wit.ie), Feb 15 2008
%F A090192 a(n) = sum_{i=1..(n-1)} q^(i-1)*a(i)*a(n-i).
%F A090192 G.f.: 1+xc(-x^2), c(x) the g.f. of A000108; a(n)=0^n+C((n-1)/2)(-1)^((n-1)/2)(1-(-1)^n)/2, where C(n)=A000108(n); - Paul Barry (pbarry(AT)wit.ie), Feb 15 2008
%Y A090192 Cf: A000108 = 1, 1, 2, 5, 14, 42, 132, 429, 1430, ...
%Y A090192 Adjacent sequences: A090189 A090190 A090191 this_sequence A090193 A090194 A090195
%Y A090192 Sequence in context: A115333 A105523 A126120 this_sequence A097331 A094032 A117780
%K A090192 sign
%O A090192 1,6
%A A090192 DELEHAM Philippe (kolotoko(AT)wanadoo.fr) Jan 22, 2004

%I A097331
%S A097331 1,1,0,1,0,2,0,5,0,14,0,42,0,132,0,429,0,1430,0,4862,0,16796,0,58786,0,
%T A097331 208012,0,742900,0,2674440,0,9694845,0,35357670,0,129644790,0,477638700,
%U A097331 0,1767263190,0,6564120420,0,24466267020,0,91482563640,0,343059613650,0
%N A097331 Expansion of 1+2x/(1+sqrt(1-4x^2)).
%C A097331 Binomial transform is A097332. Second binomial transform is A014318.
%F A097331 a(n)=0^n+Catalan((n-1)/2)(1-(-1)^n)/2
%F A097331 Unsigned version of A090192, A105523 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 29 2006
%Y A097331 Adjacent sequences: A097328 A097329 A097330 this_sequence A097332 A097333 A097334
%Y A097331 Sequence in context: A105523 A126120 A090192 this_sequence A094032 A117780 A082974
%K A097331 easy,nonn
%O A097331 0,6
%A A097331 Paul Barry (pbarry(AT)wit.ie), Aug 05 2004

%I A094032
%S A094032 0,0,2,0,5,0,15,0,44,0,129,0,407,0,1349,0,4638,0,16425,0
%N A094032 Number of n-crossing 3 component links with alternating braids of 3 strands.
%D A094032 V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math., 126 (1987), pp. 335-388.
%D A094032 K. Murasugi and B. J. Kurpita, A Study of Braids, Kluwer Academic Publishers (1999), pp. 127-166
%H A094032 T. A. Gittings, <a href="http://www.arxiv.org/abs/math.GT/0401051">Minimum braids: a complete invariant of knots and links</a>.
%Y A094032 Cf. A094029, A094030, A094031.
%Y A094032 Adjacent sequences: A094029 A094030 A094031 this_sequence A094033 A094034 A094035
%Y A094032 Sequence in context: A126120 A090192 A097331 this_sequence A117780 A082974 A066283
%K A094032 nonn
%O A094032 4,3
%A A094032 Thomas A. Gittings (tomgittings(AT)aol.com), Apr 22 2004

%I A117780
%S A117780 2,0,5,0,18,0,58,0,160,0
%N A117780 Total number of palindromic primes in base 5 with n digits.
%C A117780 Every palindrome with an even number of digits is divisible by 11 (in base 5) and therefore is composite (not prime). Hence there is no palindromic prime with an even number of digits.
%H A117780 Eric Weisstein: <a href="http://mathworld.wolfram.com/PalindromicPrime.html">Palindromic Prime</a>.
%Y A117780 Cf. A029973, A117700.
%Y A117780 Adjacent sequences: A117777 A117778 A117779 this_sequence A117781 A117782 A117783
%Y A117780 Sequence in context: A090192 A097331 A094032 this_sequence A082974 A066283 A014842
%K A117780 nonn
%O A117780 1,1
%A A117780 Martin Renner (martin.renner(AT)gmx.net), Apr 15 2006

%I A082974
%S A082974 2,0,5,1,12,8,6,2,25,23,17,13,11,7,1,54,52,46,42,40,34,30,24,16,12,10,6,
%T A082974 4,0,113,109,103,101,91,89,83,77,73,67,61,59,49,47,43,41,29,17,13,11,7,
%U A082974 1,240,230,224,218,212,210,204,200,198,188,174,170,168,164,150,144,134
%N A082974 a(n) = a(n-1) + p(n) mod p(n+1).
%C A082974 Differences when decreasing are essentially A001223, so increases occur when primes being used are roughly double those at previous increase; e.g. a(3352)=(12+31123)mod 31139=31135 and a(6257)=(1+62273)mod 62297=62274 - Henry Bottomley (se16(AT)btinternet.com), Jul 13 2003
%e A082974 a(4)=(((2%3 + 3)%5 + 5)%7 + 7)%11 = (((2+3)%5+5)%7+7)%11
%e A082974 = (((0+5)%7+7)%11 = (5+7)%11 = 1
%o A082974 (PARI) ps=0; pc=1; while (pc<100,ps+=prime(pc); ps%=prime(pc++); print1(ps","))
%Y A082974 Cf. A000040, A001223, A071089.
%Y A082974 Adjacent sequences: A082971 A082972 A082973 this_sequence A082975 A082976 A082977
%Y A082974 Sequence in context: A097331 A094032 A117780 this_sequence A066283 A014842 A132816
%K A082974 nonn
%O A082974 1,1
%A A082974 Jon Perry (perry(AT)globalnet.co.uk), May 28 2003
%E A082974 Edited by Henry Bottomley (se16(AT)btinternet.com), Jul 13 2003

%I A066283
%S A066283 0,0,0,0,0,1,0,0,0,2,0,5,2,4
%N A066283 Number of 3-dimensional polyominoes (or polycubes) with n cells and symmetry group of order exactly 6.
%Y A066283 See A000162.
%Y A066283 Adjacent sequences: A066280 A066281 A066282 this_sequence A066284 A066285 A066286
%Y A066283 Sequence in context: A094032 A117780 A082974 this_sequence A014842 A132816 A077453
%K A066283 nonn
%O A066283 1,10
%A A066283 Brendan Owen (brendan_owen(AT)yahoo.com), Jan 01 2002

%I A014842
%S A014842 2,0,5,2,9,3,7,6,15,2,21,14,19,9,25,7,31,12,27,28,42,10,38,34,35,22,
%T A014842 55,16,59,27,49,48,54,10,71,52,61,30,82,34,88,56,66,75,103,27,88,59,
%U A014842 84,64,112,46,97,56,105,96,130,28,138,114,108,70,118,66,146,94,121,86
%N A014842 Difference between A014837 and A014841.
%Y A014842 Adjacent sequences: A014839 A014840 A014841 this_sequence A014843 A014844 A014845
%Y A014842 Sequence in context: A117780 A082974 A066283 this_sequence A132816 A077453 A021491
%K A014842 nonn
%O A014842 3,1
%A A014842 Olivier Gerard (ogerard(AT)ext.jussieu.fr)

%I A132816
%S A132816 1,1,2,0,5,3,0,3,15,4,0,0,22,34,5,0,0,10,90,65,6,0,0,0,95,270,111,7,0,0,
%T A132816 0,35,490,665,175,8,0,0,0,0,406,1820,1428,260,9,0,0,0,0,126,2520,5460,
%U A132816 27772,369,10
%N A132816 A007318^(-1) * A132812.
%C A132816 Row sums = A025566 starting (1, 3, 8, 22, 61, 171, 483,...).
%F A132816 Inverse binomial transform of A132812
%e A132816 First few rows of the triangle are:
%e A132816 1;
%e A132816 1, 2;
%e A132816 0, 5, 3;
%e A132816 0, 3, 15, 4;
%e A132816 0, 0, 22, 34, 5;
%e A132816 0, 0, 10, 90, 65, 6;
%e A132816 0, 0, 0, 95, 270, 111, 7;
%e A132816 ...
%Y A132816 Cf. A132812, A025566.
%Y A132816 Adjacent sequences: A132813 A132814 A132815 this_sequence A132817 A132818 A132819
%Y A132816 Sequence in context: A082974 A066283 A014842 this_sequence A077453 A021491 A121705
%K A132816 nonn,tabl
%O A132816 0,3
%A A132816 Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 01 2007

%I A077453
%S A077453 2,0,5,3,1,9,8,7,3,2,7,7,4,4,7,5,6,1,0,1,2,6,1,2,8,6,3,1,0,9,4,1,4,
%T A077453 5,3,4,7,3,8,3,6,1,3,4,5,0,6,6,9,4,3,9,1,5,1,6,5,2,6,1,0,3,0,4,3,6,
%U A077453 2,0,9,5,0,2,7,9,8,8,9,1,3,2,6,6,9,0,7,1,4,6,4,9,0,6,3,4,0,7,7,9,5
%N A077453 Decimal expansion of 1+sqrt(11)*(sqrt(29)+sqrt(5))/24.
%H A077453 I. Jensen, <a href="http://arXiv.org/abs/cond-mat/0008178">A transfer matrix approach to the enumeration of plane meanders</a>, J. Phys. A 33 (2000), no. 34, 5953-5963.
%e A077453 2.05319873277447561012612863...
%Y A077453 Cf. A077452.
%Y A077453 Adjacent sequences: A077450 A077451 A077452 this_sequence A077454 A077455 A077456
%Y A077453 Sequence in context: A066283 A014842 A132816 this_sequence A021491 A121705 A071782
%K A077453 nonn,cons
%O A077453 1,1
%A A077453 njas, Dec 01 2002

%I A021491
%S A021491 0,0,2,0,5,3,3,8,8,0,9,0,3,4,9,0,7,5,9,7,5,3,5,9,3,4,2,9,1,5,8,1,1,
%T A021491 0,8,8,2,9,5,6,8,7,8,8,5,0,1,0,2,6,6,9,4,0,4,5,1,7,4,5,3,7,9,8,7,6,
%U A021491 7,9,6,7,1,4,5,7,9,0,5,5,4,4,1,4,7,8,4,3,9,4,2,5,0,5,1,3,3,4,7,0,2
%N A021491 Decimal expansion of 1/487.
%Y A021491 Adjacent sequences: A021488 A021489 A021490 this_sequence A021492 A021493 A021494
%Y A021491 Sequence in context: A014842 A132816 A077453 this_sequence A121705 A071782 A107363
%K A021491 nonn,cons
%O A021491 0,3
%A A021491 njas

%I A121705
%S A121705 0,1,1,2,0,5,3,4,2,11,5,10,0,25,7,24,15,20,10,55,25,50,38,41,0,125,35,
%T A121705 120,44,117,75,100,29,278,50,275,125,250,190,205,0,625,175,600,220,585,
%U A121705 336,527,375,500,145,1390,250,1375,625,1250,718,1199,950,1025,0,3125
%N A121705 triangle read by rows: 5^n expressed as the sum of two squares.
%e A121705 5^n expressed as the sum of two squares: 5^n=x^2+y^2, 0=<x<y.
%e A121705 Number of solutions for n=0,1,..: a(n)=1,1,2,2,3,3,4,4,5,5,6,6,...
%e A121705 Triangle of solutions for n=0,1,..:
%e A121705 {x,y}
%e A121705 {{0,1}},
%e A121705 {{1,2}},
%e A121705 {{0,5},{3,4}},
%e A121705 {{2,11},{5,10}},
%e A121705 {{0,25},{7,24},{15,20}},
%e A121705 {{10,55},{25,50},{38,41}},
%e A121705 {{0,125},{35,120},{44,117},{75,100}},
%e A121705 {{29,278},{50,275},{125,250},{190,205}},
%e A121705 {{0,625},{175,600},{220,585},{336,527},{375,500}},
%e A121705 {{145,1390},{250,1375},{625,1250},{718,1199},{950,1025}},
%e A121705 {{0,3125},{237,3116},{875,3000},{1100,2925},{1680,2635},{1875,2500}},
%e A121705 {{725,6950},{1250,6875},{2642,6469},{3125,6250},{3590,5995},{4750,5125}},
%e A121705 {{0,15625},{1185,15580},{4375,15000},{5500,14625},{8400,13175},{9375,12500},10296,11753}}
%Y A121705 Cf. A004431, A006496, A024507, A120905.
%Y A121705 Adjacent sequences: A121702 A121703 A121704 this_sequence A121706 A121707 A121708
%Y A121705 Sequence in context: A132816 A077453 A021491 this_sequence A071782 A107363 A095245
%K A121705 nonn,tabl
%O A121705 0,4
%A A121705 Zak Seidov (zakseidov(AT)yahoo.com), Sep 10 2006

%I A071782
%S A071782 0,1,1,1,0,2,0,5,3,5,0,2,0,0,0,14,0,6,0,15,7,0,0,18,0,13,18,0,0,0,0,
%T A071782 8,0,17,0,24,0,0,13,15,0,14,0,22,0,0,0,28,0,25,0,39,0,9,0,28,19,29,0,
%U A071782 30,0,0,21,32,0,0,0,17,0,0,0,36,0,37,50,38,0,26,0,10
%N A071782 Sum of distinct squares in Z_n (mod n).
%C A071782 a(n)=0 for prime n>3. - T. D. Noe (noe(AT)sspectra.com), Sep 06 2005
%t A071782 a[n_] := Mod[Apply[Plus, Union[Table[Mod[i^2, n], {i, 1, n}]]], n]
%Y A071782 Adjacent sequences: A071779 A071780 A071781 this_sequence A071783 A071784 A071785
%Y A071782 Sequence in context: A077453 A021491 A121705 this_sequence A107363 A095245 A086280
%K A071782 nonn
%O A071782 1,6
%A A071782 Santi Spadaro (spados(AT)katamail.com), Jun 24 2002

%I A107363
%S A107363 1,1,1,1,2,0,5,3,7,3,8,0,21,13,29,13,34,0,89,55,123,55,144,0,377,233,521,233,610,
%T A107363 0,1597,987,2207,987,2584,0,6765,4181,9349,4181,10946,0,28657,17711,39603,17711,
%U A107363 46368,0,121393,75025,167761,75025,196418,0,514229,317811,710647,317811,832040,0
%V A107363 1,1,-1,1,2,0,5,3,-7,3,8,0,21,13,-29,13,34,0,89,55,-123,55,144,0,377,233,-521,233,610,
%W A107363 0,1597,987,-2207,987,2584,0,6765,4181,-9349,4181,10946,0,28657,17711,-39603,17711,
%X A107363 46368,0,121393,75025,-167761,75025,196418,0,514229,317811,-710647,317811,832040,0
%N A107363 G.f. (x-1)*(1+x^2)*(x^4+2*x^3-x^2+1)*(x+1)^2/((x^4+x^2-1)*(x^8-x^6+2*x^4+x^2+1)).
%C A107363 Conjectures: { Fib(n) | n in naturals } = { a(n) | n in naturals, a(n) >= 0 } = { a(n) | n in naturals, n not of the form 6*n+2 } (naturals include 0).
%F A107363 a(6*n+2) = - A048876(n) (Generalized Pellian with second term of 7), conjecture
%o A107363 Floretion Algebra Multiplication Program, FAMP Code: 4teszapseq[(- .5'j + .5'k - .5j' + .5k' - 'ii' - .5'ij' - .5'ik' - .5'ji' - .5'ki')*( + .5'j + .5i' + .5'ik' + .5'jk' + .5'ki' + .5'kj')]
%Y A107363 Cf. A000045, A048876.
%Y A107363 Adjacent sequences: A107360 A107361 A107362 this_sequence A107364 A107365 A107366
%Y A107363 Sequence in context: A021491 A121705 A071782 this_sequence A095245 A086280 A083714
%K A107363 sign
%O A107363 0,5
%A A107363 Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), May 24 2005

%I A095245
%S A095245 0,2,0,5,3,7,15,16,21,27,7,11,17,21,37,7,28,10,50,70,70,23,46,20,76,93,
%T A095245 81,52,1,58,87,54,100,128,39,10,117,16,42,89,98,61,135,123,13,89,201,
%U A095245 147,124,176,186,202,71,74,256,228,137,84,145
%N A095245 (Concatenation of first n primes) modulo prime(n).
%e A095245 The concatenation of the first 3 primes is 235. The third prime is 5. Therefore a(3) = 235 mod 5 = 0.
%t A095245 a = {2}; b = {0}; For[n = 2, n < 100, n++, a = Flatten[Join[a, IntegerDigits[ Prime[n]]] ]; AppendTo[b, Mod[FromDigits[a], Prime[n]]]]; b
%Y A095245 Cf. A095243.
%Y A095245 Adjacent sequences: A095242 A095243 A095244 this_sequence A095246 A095247 A095248
%Y A095245 Sequence in context: A121705 A071782 A107363 this_sequence A086280 A083714 A137421
%K A095245 base,nonn,less
%O A095245 0,2
%A A095245 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jun 17 2004
%E A095245 Edited, corrected and extended by Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Jun 26 2007

%I A086280
%S A086280 0,0,2,0,5,3,8,3,4,4,2,0,3,0,3,3,4,5,8,6,6,1,6,0,0,4,6,5,4,2,7,5,3,3,8,
%T A086280 4,2,8,5,7,1,5,8,0,4,4,4,5,4,1,0,6,1,8,2,4,5,4,8,1,4,8,3,3,3,6,9,1,3,8,
%U A086280 3,4,4,9,2,1,1,2,9,7,0,0,5,3,5,7,0,5,5,7,1,6,6,2,2,8,5,6,6,7,0,2
%N A086280 Decimal expansion of 3rd Stieltjes constant gamma_3.
%H A086280 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StieltjesConstants.html">Stieltjes Constants</a>
%e A086280 0.0020538...
%Y A086280 Cf. A082633, A086279, A086281, A086282.
%Y A086280 Adjacent sequences: A086277 A086278 A086279 this_sequence A086281 A086282 A086283
%Y A086280 Sequence in context: A071782 A107363 A095245 this_sequence A083714 A137421 A051111
%K A086280 nonn,cons
%O A086280 0,3
%A A086280 Eric Weisstein (eric(AT)weisstein.com), Jul 14, 2003

%I A083714
%S A083714 0,0,0,1,0,2,0,5,4,2,0,8,0,6,8,11,0,14,0,14,12,8,0,20,18,10,20,16,0,24,
%T A083714 0,29,20,14,24,28,0,18,24,32,0,34,0,32,38,20,0,44,40,42,30,34,0,50,42,
%U A083714 46,34,24,0,54,0,30,54,59,48,50,0,50,44,60,0,68,0,36,68,54,62,60,0,74
%N A083714 (greatest prime <= n) - (greatest prime factor of n).
%C A083714 a(n) = A007917(n) - A006530(n);
%C A083714 n>1: a(n) = 0 iff n is prime.
%Y A083714 Cf. A083718, A083715, A083716, A083717.
%Y A083714 Adjacent sequences: A083711 A083712 A083713 this_sequence A083715 A083716 A083717
%Y A083714 Sequence in context: A107363 A095245 A086280 this_sequence A137421 A051111 A068558
%K A083714 nonn
%O A083714 1,6
%A A083714 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), May 04 2003

%I A137421
%S A137421 1,2,0,5,5,6,9,4,3,0,4,0,0,5,9,0,3,1,1,7,0,2,0,2,8,6,1,7,7,8,3,8,2,3,4,
%T A137421 2,6,3,7,7,1,0,8,9,1,9,5,9,7,6,9,9,4,4,0,4,7,0,5,5,2,2,0,3,5,5,1,8,3,4,
%U A137421 7,9,0,3,5,9,1,6
%N A137421 Decimal expansion of growth constant in random Fibonacci sequence.
%H A137421 Elise Janvresse, Benoit Rittaud and Thierry De La Rue, <a href="http://arxiv.org/pdf/0804.2400">Growth rate for the expected value of a generalized random Fibonacci sequence</a>
%e A137421 alpha - 1 = 1.20556943... where alpha is the only real root of alpha^3 = 2*alpha^2 + 1. This is the growth rate of the expected value of a (1/2,1)-random Fibonacci sequence, defined by the relation g_n = | g_{n-1} +/- g_{n-2} |, where the +/- sign is chosen by tossing a balanced coin for each n. The more general case of an unbalanced coin is given by Janvresse, Rittaud and De La Rue.
%p A137421 Digits := 80 ; fsolve( x^3-2*x^2-1,x,2.2..2.3)-1.0 ; - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 23 2008
%Y A137421 Adjacent sequences: A137418 A137419 A137420 this_sequence A137422 A137423 A137424
%Y A137421 Sequence in context: A095245 A086280 A083714 this_sequence A051111 A068558 A082832
%K A137421 easy,nonn,cons,new
%O A137421 1,2
%A A137421 Jonathan Vos Post (jvospost3(AT)gmail.com), Apr 16 2008
%E A137421 More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 23 2008

%I A051111
%S A051111 1,2,0,5,5,8,21,0,55,55,89,233,0,610,610,987,2584,0,6765,6765,10946,
%T A051111 28657,0,75025,75025,121393,317811,0,832040,832040,1346269,3524578,0,
%U A051111 9227465,9227465,14930352,39088169,0,102334155,102334155
%V A051111 1,2,0,-5,-5,8,21,0,-55,-55,89,233,0,-610,-610,987,2584,0,-6765,-6765,
%W A051111 10946,28657,0,-75025,-75025,121393,317811,0,-832040,-832040,1346269,
%X A051111 3524578,0,-9227465,-9227465,14930352,39088169,0,-102334155,-102334155
%N A051111 Expansion of x/(x^4-3*x^3+4*x^2-2*x+1).
%F A051111 a(5n+1)=F(5n+1), a(5n+2)=F(5n+3), a(5n+3)=0, a(5n-1)=a(5n)=-F(5n), where F=A000045 the Fibonacci sequence.
%F A051111 G.f.: x/(x^4-3x^3+4x^2-2x+1). - Michael Somos, Apr 25 2003
%o A051111 (PARI) a(n)=local(x,y); x=fibonacci(n); y=fibonacci(n+1); [ -x,x,y,0,-y][n%5+1]
%Y A051111 Adjacent sequences: A051108 A051109 A051110 this_sequence A051112 A051113 A051114
%Y A051111 Sequence in context: A086280 A083714 A137421 this_sequence A068558 A082832 A097709
%K A051111 sign
%O A051111 1,2
%A A051111 Michael Somos

%I A068558
%S A068558 2,0,5,6,0,0,9,6,4,5,3,6,1,2,1,9,3,8,2,6,0,4,2,4,8,2,1,7,5,7,4,7,0,5,4,
%T A068558 9,5,4,0,4,2,1,4,2,4,6,3,3,2,4,2,6,2,5,9,3,5,2,3,6,1,8,9,4,7,4,0,7,0,4,
%U A068558 5,1,1,2,3,9,7,2,7,6,8,4,5,1,6,0,1,5,4,8,9,8,7,3,6,4,0,7,4,1,8,2,0,1,3
%N A068558 Decimal expansion of the smallest solution >0 to cos(x)=cos(x^2).
%C A068558 Let d(n) be defined to be the smallest solution to cos(x)=cos(x^n) then lim n -> infinity d(n) exists = C = 2,36338112904...= w004 in Plouffe's inverter.
%F A068558 This number is (-1+sqrt(1+8Pi))/2 = 2, 0560096...
%Y A068558 Adjacent sequences: A068555 A068556 A068557 this_sequence A068559 A068560 A068561
%Y A068558 Sequence in context: A083714 A137421 A051111 this_sequence A082832 A097709 A104555
%K A068558 base,easy,nonn,cons
%O A068558 1,1
%A A068558 Benoit Cloitre (benoit7848c(AT)orange.fr), Mar 30 2002

%I A082832
%S A082832 2,0,5,6,9,8,7,7,9,5,0,9,6,1,2,3,0,3,7,1,0,7
%N A082832 Decimal expansion of the (finite) value of the sum_{ k >= 1, k has no digit equal to 3 in base 10 } 1/k.
%D A082832 Robert Baillie, Sums of reciprocals of integers missing a given digit, Amer. Math. Monthly, 86 (1979), 372-374.
%D A082832 Julian Havil, Gamma, Exploring Euler's Constant, Princeton University Press, Princeton and Oxford, 2003, page 34.
%e A082832 20.56987795096123037107...
%Y A082832 Cf. A002387, A024101, A082830, A082831, A082833, A082834, A082835, A082836, A082837, A082838, A082839.
%Y A082832 Adjacent sequences: A082829 A082830 A082831 this_sequence A082833 A082834 A082835
%Y A082832 Sequence in context: A137421 A051111 A068558 this_sequence A097709 A104555 A078049
%K A082832 nonn,cons,more,base
%O A082832 2,1
%A A082832 Robert G. Wilson v (rgwv(AT)rgwv.com), Apr 14 2003

%I A097709
%S A097709 2,0,5,6,23,16,35,54,35,10,11,40,57,30,477,58,215,0,715,264,83,16,105,
%T A097709 30,379,238,5,462,851,538,1,70,199,618,85,84,881,454,53,310,223,330,
%U A097709 1109,540,2059,398,305,120,185,574,411,340,539,318,209,648,421,2444,73
%N A097709 Least k such that A097708(n)+k is prime.
%C A097709 Are there any more zeros in this sequence after a(17)?
%H A097709 Prime Curios, <a href="http://primes.utm.edu/curios/page.php?number_id=4437">11111...66787 (59-digits)</a>.
%Y A097709 Cf. A097708.
%Y A097709 Adjacent sequences: A097706 A097707 A097708 this_sequence A097710 A097711 A097712
%Y A097709 Sequence in context: A051111 A068558 A082832 this_sequence A104555 A078049 A021490
%K A097709 nonn
%O A097709 0,1
%A A097709 Jason Earls (jcearls(AT)cableone.net), Aug 21 2004

%I A104555
%S A104555 0,1,2,0,5,7,0,12,15,0,22,26,0,35,40,0,51,57,0,70,77,0,92,100,0,117,126,
%T A104555 0,145,155,0,176,187,0,210,222,0,247,260,0,287,301,0,330,345,0,376,392,
%U A104555 0,425,442,0,477,495
%V A104555 0,1,2,0,-5,-7,0,12,15,0,-22,-26,0,35,40,0,-51,-57,0,70,77,0,-92,-100,0,117,126,0,-145,
%W A104555 -155,0,176,187,0,-210,-222,0,247,260,0,-287,-301,0,330,345,0,-376,-392,0,425,442,0,
%X A104555 -477,-495
%N A104555 Expansion of x(1-x)/(1-x+x^2)^3.
%C A104555 Image of C(n+1,2) under the Riordan array (1, x(1-x)).
%F A104555 a(n)=3a(n-1)-6a(n-2)+7a(n-3)-6a(n-4)+3a(n-5)-a(n-6); a(n)=sum{k=0..n, binomial(k, n-k)(-1)^(n-k)*k(k+1)/2}; a(n)=sum{k=0..floor(n/2), binomial(n-k, k)(-1)^k*(n-k)(n-k+1)/2}.
%F A104555 a(3n) = 0, a(3n-2) = (3n^2-n)/2, a(3n-1) = (3n^2+n)/2. - Ralf Stephan, May 20 2007
%Y A104555 Adjacent sequences: A104552 A104553 A104554 this_sequence A104556 A104557 A104558
%Y A104555 Sequence in context: A068558 A082832 A097709 this_sequence A078049 A021490 A084258
%K A104555 easy,sign
%O A104555 0,3
%A A104555 Paul Barry (pbarry(AT)wit.ie), Mar 14 2005

%I A078049
%S A078049 1,2,0,5,7,3,22,23,24,92,67,141,367,152,723,1394,100,3411,5005,1717,15138,
%T A078049 16709,15284,63840,49981,92983,256785,120800,485753,984138,133432,2320597,
%U A078049 3571599,936163,10399958,12099231,9636848,44235268,37060803,61046581,179403455
%V A078049 1,-2,0,5,-7,-3,22,-23,-24,92,-67,-141,367,-152,-723,1394,-100,-3411,5005,1717,-15138,
%W A078049 16709,15284,-63840,49981,92983,-256785,120800,485753,-984138,133432,2320597,
%X A078049 -3571599,-936163,10399958,-12099231,-9636848,44235268,-37060803,-61046581,179403455
%N A078049 Expansion of (1-x)/(1+x+2*x^2-x^3).
%Y A078049 Cf. A077978.
%Y A078049 Adjacent sequences: A078046 A078047 A078048 this_sequence A078050 A078051 A078052
%Y A078049 Sequence in context: A082832 A097709 A104555 this_sequence A021490 A084258 A111352
%K A078049 sign
%O A078049 0,2
%A A078049 njas, Nov 17 2002

%I A021490
%S A021490 0,0,2,0,5,7,6,1,3,1,6,8,7,2,4,2,7,9,8,3,5,3,9,0,9,4,6,5,0,2,0,5,7,
%T A021490 6,1,3,1,6,8,7,2,4,2,7,9,8,3,5,3,9,0,9,4,6,5,0,2,0,5,7,6,1,3,1,6,8,
%U A021490 7,2,4,2,7,9,8,3,5,3,9,0,9,4,6,5,0,2,0,5,7,6,1,3,1,6,8,7,2,4,2,7,9
%N A021490 Decimal expansion of 1/486.
%Y A021490 Adjacent sequences: A021487 A021488 A021489 this_sequence A021491 A021492 A021493
%Y A021490 Sequence in context: A097709 A104555 A078049 this_sequence A084258 A111352 A133446
%K A021490 nonn,cons
%O A021490 0,3
%A A021490 njas

%I A084258
%S A084258 1,2,0,5,7,9,9,6,4,8,6,7,8,3,2,6,3,4,0,1,5,7,4,1,2,2,5,2,6,0,9,4,9,8,7,0,
%T A084258 2,3,0,8,7,6,1,2,2,2,0,0,6,6,4,3,0,7,6,9,9,4,5,0,9,8,1,5,1,4,8,0,2,6,4,6,
%U A084258 9,0,1,2,5,5,5,2,3,4,7,9,4,2,6,0,5,9,5,7,1,2,3,3,4,4,6,3,0,6,2,2,8,2,5,2
%N A084258 Decimal expansion of c=sum(k>=1, coth(Pi*k)/k^3 ).
%C A084258 Splitting the infinite sum Simon Plouffe unearthed a rapidly converging series for zeta(3).
%D A084258 Bruce C. Berndt, Ramanujan Notebook part II, Infinite series, Springer Verlag, p. 293.
%H A084258 Simon Plouffe, <a href="http://www.lacim.uqam.ca/~plouffe/articles/Identities.pdf">Formulae for zeta(2n+1)</a>.
%F A084258 c=7*Pi^3/180
%o A084258 (PARI) 7*Pi^3/180
%Y A084258 Adjacent sequences: A084255 A084256 A084257 this_sequence A084259 A084260 A084261
%Y A084258 Sequence in context: A104555 A078049 A021490 this_sequence A111352 A133446 A011122
%K A084258 cons,nonn
%O A084258 1,2
%A A084258 Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 21 2003

%I A111352
%S A111352 1,1,2,0,5,7,22,48,121,287,698,1680,4061,9799,23662,57120,137905,332927,
%T A111352 803762,1940448,4684661,11309767,27304198,65918160,159140521,384199199,
%U A111352 927538922,2239277040,5406093005
%V A111352 1,-1,2,0,5,7,22,48,121,287,698,1680,4061,9799,23662,57120,137905,332927,803762,
%W A111352 1940448,4684661,11309767,27304198,65918160,159140521,384199199,927538922,2239277040,
%X A111352 5406093005
%N A111352 a(n+3) = a(n+2) + 3*a(n+1) + a(n).
%C A111352 a(n) + a(n+1) = A000129(n); a(n+2) - a(n) = A001333(n)
%D A111352 C. Dement, Floretion-generated Integer Sequences (work in progress)
%F A111352 a(n) = (1/4*sqrt(2)-1/4)*(1+sqrt(2))^n + (-1/4*sqrt(2)-1/4)*(1-sqrt(2))^n + 3/2*(-1)^n; G.f. (2*x-1)/((x+1)*(x^2+2*x-1))
%o A111352 Floretion Algebra Multiplication Program, FAMP Code: 2jbasekrokseq[A*H] with A = + .25'i + .25i' + .25'ii' + .25'jj' + .25'kk' + .25'jk' + .25'kj' + .25e; H = - 2'i - 'j + 'k; roktype : Y[15] = Y[15] + (-1)^p (internal program code)
%Y A111352 Cf. A000129, A001333.
%Y A111352 Adjacent sequences: A111349 A111350 A111351 this_sequence A111353 A111354 A111355
%Y A111352 Sequence in context: A078049 A021490 A084258 this_sequence A133446 A011122 A085009
%K A111352 easy,sign
%O A111352 0,3
%A A111352 Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Oct 29 2005

%I A133446
%S A133446 2,0,5,8,2,9,5,6,0,8
%N A133446 Decimal expansion of the number c such that the solution to the differential functional equation f'(x) = f(x-1) + f(x-2) is c^x.
%C A133446 This is related to phi, as phi^x is the solution to f(x) = f(x-1) + f(x-2).
%F A133446 f(x-1) + f(x-2) = f'(x), f(x) = 2.058295608^x
%p A133446 solve(ln(x)*x^2=x+1)
%Y A133446 Adjacent sequences: A133443 A133444 A133445 this_sequence A133447 A133448 A133449
%Y A133446 Sequence in context: A021490 A084258 A111352 this_sequence A011122 A085009 A011435
%K A133446 cons,nonn
%O A133446 1,1
%A A133446 Cameron Davidson-Pilon (see_dee_pee(AT)hotmail.com), Nov 26 2007

%I A011122
%S A011122 2,0,5,8,9,2,4,1,3,6,4,7,8,5,1,7,2,5,2,4,6,0,0,3,0,4,1,6,0,6,6,1,8,
%T A011122 6,5,8,6,9,1,8,1,3,3,7,9,9,4,6,4,7,2,2,0,0,0,6,1,4,6,7,4,0,0,3,1,1,
%U A011122 2,0,9,0,6,6,0,3,9,8,6,0,9,1,8,2,2,5,1,8,5,6,1,3,6,7,7,9,1,1,5,1,9
%N A011122 Decimal expansion of 5th root of 37.
%Y A011122 Adjacent sequences: A011119 A011120 A011121 this_sequence A011123 A011124 A011125
%Y A011122 Sequence in context: A084258 A111352 A133446 this_sequence A085009 A011435 A011014
%K A011122 nonn,cons
%O A011122 1,1
%A A011122 njas

%I A085009
%S A085009 2,0,5,8,9,8,5,0,2,2,0,5,8,9,17,23,27,29,29,27,32,35,36,35,32,27,29,29,
%T A085009 27,23,17,9,8,5,0,2,2,0,5,8,9,8,5,0,2,2,0,5,8,9,17,23,27,29,29,27,32,35,
%U A085009 36,44,50,54,65,74,81,86,89,90,98,104,108,110,110,108,113,116,117,116
%N A085009 "Von Koch" sequence generated by {1,1,2}.
%C A085009 The graph of the sequence is similar to, for example, A071992
%F A085009 a(n)= n+2 + sum(k=1, n, A085008(k))
%Y A085009 Cf. A085006, A085007, A085009, A005536 ("Von Koch" sequence generated by {1, 2}).
%Y A085009 Adjacent sequences: A085006 A085007 A085008 this_sequence A085010 A085011 A085012
%Y A085009 Sequence in context: A111352 A133446 A011122 this_sequence A011435 A011014 A002976
%K A085009 nonn
%O A085009 1,1
%A A085009 Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 17 2003

%I A011435
%S A011435 1,2,0,5,9,0,8,5,5,1,0,3,0,6,9,6,3,9,1,6,2,1,8,9,7,2,0,0,7,6,7,7,9,
%T A011435 3,0,1,8,0,8,6,5,9,4,3,7,4,9,2,3,1,1,7,4,5,6,7,2,2,6,3,8,9,4,8,4,9,
%U A011435 1,6,7,6,9,5,8,6,1,7,0,8,0,9,4,6,6,0,9,0,8,3,9,0,3,8,2,8,9,0,5,8,5
%N A011435 Decimal expansion of 16th root of 20.
%Y A011435 Adjacent sequences: A011432 A011433 A011434 this_sequence A011436 A011437 A011438
%Y A011435 Sequence in context: A133446 A011122 A085009 this_sequence A011014 A002976 A080901
%K A011435 nonn,cons
%O A011435 1,2
%A A011435 njas

%I A011014
%S A011014 2,0,5,9,7,6,7,1,4,3,9,0,7,1,1,7,7,5,5,8,3,0,2,7,7,2,5,5,2,0,1,0,1,
%T A011014 0,7,8,0,1,0,2,6,9,2,7,4,4,1,5,5,4,0,4,0,7,8,5,9,7,2,0,1,4,4,6,6,2,
%U A011014 3,2,5,5,0,5,0,0,2,3,7,4,9,8,5,6,3,7,6,9,2,1,0,5,1,3,7,1,3,7,2,5,2
%N A011014 Decimal expansion of 4th root of 18.
%Y A011014 Adjacent sequences: A011011 A011012 A011013 this_sequence A011015 A011016 A011017
%Y A011014 Sequence in context: A011122 A085009 A011435 this_sequence A002976 A080901 A137260
%K A011014 nonn,cons
%O A011014 1,1
%A A011014 njas

%I A002976 M0034
%S A002976 0,1,0,2,0,5,9,21,42,76,174,396,888,2023,4345,9921,22566
%N A002976 Number of self-avoiding walks with n steps on square lattice.
%D A002976 W. A. Beyer and M. B. Wells, Lower bound for the connective constant of a self-avoiding walk on a square lattice, J. Combin. Theory, A 13 (1972), 176-182.
%F A002976 a(n)=A006142(n)+2*A006143(n)+A006144(n). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 22 2007
%Y A002976 Adjacent sequences: A002973 A002974 A002975 this_sequence A002977 A002978 A002979
%Y A002976 Sequence in context: A085009 A011435 A011014 this_sequence A080901 A137260 A047918
%K A002976 nonn,walk,more
%O A002976 4,4
%A A002976 njas

%I A080901
%S A080901 2,0,5,10,8,13,18,16,21,19,24,29,27,32,37,35,40,38,36,41,39,44,49,
%T A080901 47,52,57,55,60,58,63,68,66,71,76,74,72,70,68,66,64,62,67,72,70,75,
%U A080901 80,78,83,81,86,91,89,94,99,97,102,100,98,103,101,106,104,102,100
%N A080901 a(1)=2; for n>1, a(n)=a(n-1)-2 if n is already in the sequence, a(n)=a(n-1)+5 otherwise.
%H A080901 B. Cloitre, N. J. A. Sloane and M. J. Vandermast, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Numerical analogues of Aronson's sequence</a>, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
%H A080901 B. Cloitre, N. J. A. Sloane and M. J. Vandermast, <a href="http://arXiv.org/abs/math.NT/0305308">Numerical analogues of Aronson's sequence</a> (math.NT/0305308)
%F A080901 a(n)-n (n >= 1) (A081745) is periodic with period 168.
%Y A080901 Cf. A080578, A080900, A081745.
%Y A080901 Adjacent sequences: A080898 A080899 A080900 this_sequence A080902 A080903 A080904
%Y A080901 Sequence in context: A011435 A011014 A002976 this_sequence A137260 A047918 A138701
%K A080901 nonn,easy
%O A080901 1,1
%A A080901 njas and Benoit Cloitre, Apr 01 2003

%I A137260
%S A137260 0,0,0,1,2,0,5,10,12,0,23,46,66,72,0,119,238,354,456,480,0,719,1438,
%T A137260 2154,2856,3480,3600,0,5039,10078,15114,20136,25080,29520,30240,0,40319,
%U A137260 80638,120954,161256,201480,241200,277200,282240,0,362879,725758
%N A137260 Triangular sequence of limited permutations of the form: t(n,m)=-n!+n*(m-1)!.
%C A137260 Inspired by the formula: Sum[k*p[n,k],{k,0,n}]=n!
%D A137260 http://www.maa.org/pubs/monthly_mar08_toc.html The Fubini Principle By: Krassimir Penev krassi(AT)att.net
%F A137260 t(n,m)=n!-n*(m-1)!
%e A137260 {0},
%e A137260 {0, 0},
%e A137260 {1, 2, 0},
%e A137260 {5, 10, 12, 0},
%e A137260 {23, 46, 66, 72, 0},
%e A137260 {119, 238, 354, 456, 480, 0},
%e A137260 {719, 1438, 2154, 2856, 3480, 3600, 0},
%e A137260 {5039, 10078, 15114, 20136, 25080, 29520, 30240, 0},
%e A137260 {40319, 80638, 120954, 161256, 201480, 241200, 277200, 282240, 0},
%e A137260 {362879, 725758, 1088634, 1451496, 1814280, 2176560, 2535120, 2862720, 2903040, 0}
%t A137260 t[n_, m_] = -n! + n*(m - 1)!; a = Table[Table[t[n, m], {n, 1, m}], {m, 1, 10}]; Flatten[a]
%Y A137260 Adjacent sequences: A137257 A137258 A137259 this_sequence A137261 A137262 A137263
%Y A137260 Sequence in context: A011014 A002976 A080901 this_sequence A047918 A138701 A050821
%K A137260 nonn,uned
%O A137260 1,5
%A A137260 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Mar 11 2008

%I A047918
%S A047918 1,2,0,6,0,0,8,0,0,16,20,0,0,0,100,12,24,36,0,0,648,42,0,0,0,0,
%T A047918 0,4998,32,32,0,320,0,0,0,39936,54,0,270,0,0,0,0,0,362556,40,160,
%U A047918 0,0,3800,0,0,0,0,3624800,110,0,0,0,0,0,0,0,0,0,39916690,48,96
%N A047918 Triangular array a(n,k) = Sum_{d|k} mu(d)*U(n,k/d) if k|n else 0, where U(n,k)=A047916(n,k) (1<=k<=n).
%D A047918 J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61.
%Y A047918 Adjacent sequences: A047915 A047916 A047917 this_sequence A047919 A047920 A047921
%Y A047918 Sequence in context: A002976 A080901 A137260 this_sequence A138701 A050821 A076257
%K A047918 nonn,tabl,nice,easy
%O A047918 0,2
%A A047918 njas

%I A138701
%S A138701 1,0,2,0,6,0,0,30,0,0,42,0,0,30,0,0,13,5,0,0,3,1,19,3,11,0,1,6,0
%N A138701 Irregular array read by rows: row n contains the continued fraction terms (in order) for the absolute value of B_n, the nth Bernoulli number.
%C A138701 Row n, for all odd n >= 3, is (0).
%C A138701 The number of terms in row n is A138702(n).
%e A138701 The 12th Bernoulli number is -691/2730. Now 691/2730 has the continued fraction 0 + 1/(3 + 1/(1 + 1/(19 + 1/(3 + 1/11)))). So row 12 is (0,3,1,19,3,11).
%Y A138701 Cf. A138702, A138703, A138704, A027641, A027642.
%Y A138701 Adjacent sequences: A138698 A138699 A138700 this_sequence A138702 A138703 A138704
%Y A138701 Sequence in context: A080901 A137260 A047918 this_sequence A050821 A076257 A046520
%K A138701 more,nonn,tabf
%O A138701 0,3
%A A138701 Leroy Quet (qq-quet(AT)mindspring.com), Mar 26 2008

%I A050821
%S A050821 0,2,0,6,0,2,8,6,8,10,10,8,14,0,0,26,4,18,8,14,18,20,20,18,28,14,12,26,
%T A050821 44,14,26,24,38,16,10,32,22,32,64,10,36,38,22,36,64,10,24,14,6,24,38,
%U A050821 44,74,28,36,50,92,14,52,50,46,56,72,38,24,106,6,68,30,0,80,58,42,20,4
%N A050821 Even numbers in the sequence generated by a(n)=|a(n-1)+2a(n-2)-n|.
%Y A050821 Cf. A005210.
%Y A050821 Equals 2*A039727(n).
%Y A050821 Adjacent sequences: A050818 A050819 A050820 this_sequence A050822 A050823 A050824
%Y A050821 Sequence in context: A137260 A047918 A138701 this_sequence A076257 A046520 A019781
%K A050821 nonn,easy
%O A050821 1,2
%A A050821 Mohammad K. Azarian, azarian(AT)evansville.edu
%E A050821 More terms from James A. Sellers (sellersj(AT)math.psu.edu)

%I A076257
%S A076257 1,2,0,6,0,2,24,0,24,0,120,0,240,0,24,720,0,2400,0,720,0,5040,0,25200,
%T A076257 0,15120,0,720,40320,0,282240,0,282240,0,40320,0,362880,0,3386880,0,
%U A076257 5080320,0,1451520,0,40320,3628800,0,43545600,0,91445760,0,43545600,0
%V A076257 1,-2,0,6,0,-2,-24,0,24,0,120,0,-240,0,24,-720,0,2400,0,-720,0,5040,0,-25200,0,15120,
%W A076257 0,-720,-40320,0,282240,0,-282240,0,40320,0,362880,0,-3386880,0,5080320,0,-1451520,0,
%X A076257 40320,-3628800,0,43545600,0,-91445760,0,43545600,0
%N A076257 Coefficients of the polynomials in the numerator of 1/(1+x^2) and its successive derivatives, starting with the coefficient of the highest power of x.
%C A076257 Denominator of n-th derivative is (1+x^2)^(n+1), whose coefficients are the binomial coefficients, A007318.
%F A076257 For 0<=k<=n, let a(n, k) be the coefficient of x^k in the numerator of the n-th derivative of 1/(1+x^2). If n+k is even, a(n, k) = (-1)^((n+k)/2)*n!*binomial(n+1, k); if n+k is odd, a(n, k)=0.
%e A076257 The coefficients of the numerators starting with the coefficient of the highest power of x are 1; -2,0; 6,0,-2; -24,0,24,0; ...
%t A076257 a[n_, k_] := Coefficient[Expand[Together[(1+x^2)^(n+1)*D[1/(1+x^2), {x, n}]]], x, k]; Flatten[Table[a[n, k], {n, 0, 10}, {k, n, 0, -1}]]
%Y A076257 Cf. A076256, A076741, A076743.
%Y A076257 Adjacent sequences: A076254 A076255 A076256 this_sequence A076258 A076259 A076260
%Y A076257 Sequence in context: A047918 A138701 A050821 this_sequence A046520 A019781 A081153
%K A076257 sign,tabl,easy
%O A076257 0,2
%A A076257 Mohammad K. Azarian (azarian(AT)evansville.edu), Nov 05 2002
%E A076257 Edited by Dean Hickerson (dean(AT)math.ucdavis.edu), Nov 28 2002

%I A046520
%S A046520 1,0,0,2,0,6,0,7,4,10,0,18,0,14,12,18,0,27,0,28,16,22,0,44,8,26,18,38,0,
%T A046520 56,0,41,24,34,20,70,0,38,28,66,0,76,0,58,48,46,0,98,12,67,36,68,0,94,
%U A046520 28,88,40,58,0,140,0,62,62,88,32,116,0,88,48,112,0,159,0,74
%V A046520 -1,0,0,2,0,6,0,7,4,10,0,18,0,14,12,18,0,27,0,28,16,22,0,44,8,26,18,38,0,56,0,41,24,34,
%W A046520 20,70,0,38,28,66,0,76,0,58,48,46,0,98,12,67,36,68,0,94,28,88,40,58,0,140,0,62,62,88,
%X A046520 32,116,0,88,48,112,0,159,0,74
%N A046520 Sum of divisors of n - phi(n) - number of divisors of n.
%C A046520 Always >= 0 for n >= 2. a(n)=0 if and only if n is prime.
%D A046520 D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section I.3.1 (but they have "tau" instead of "sigma").
%H A046520 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b046520.txt">Table of n, a(n) for n=1..10000</a>
%H A046520 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/sg.txt">My favorite integer sequences</a>, in Sequences and their Applications (Proceedings of SETA '98).
%Y A046520 Cf. A079538, A079539
%Y A046520 a(n) = A000203(n) - A000010(n) - A000005(n).
%Y A046520 Adjacent sequences: A046517 A046518 A046519 this_sequence A046521 A046522 A046523
%Y A046520 Sequence in context: A138701 A050821 A076257 this_sequence A019781 A081153 A126869
%K A046520 sign,easy
%O A046520 1,4
%A A046520 njas
%E A046520 Corrected by Dean Hickerson, dean(AT)math.ucdavis.edu, Dec 19 2006

%I A019781
%S A019781 2,0,6,0,9,0,1,5,8,8,3,7,5,1,6,0,1,8,3,5,6,0,8,1,3,4,8,4,7,6,7,7,0,
%T A019781 4,4,6,4,5,6,7,2,2,0,1,2,5,8,8,7,6,8,5,0,1,4,4,6,8,8,4,9,1,3,9,5,5,
%U A019781 0,8,2,6,2,7,6,4,9,2,7,6,9,5,1,0,7,9,0,9,7,0,6,5,0,0,7,0,4,5,8,3,0
%N A019781 Decimal expansion of sqrt(E)/8.
%Y A019781 Adjacent sequences: A019778 A019779 A019780 this_sequence A019782 A019783 A019784
%Y A019781 Sequence in context: A050821 A076257 A046520 this_sequence A081153 A126869 A094233
%K A019781 nonn,cons
%O A019781 0,1
%A A019781 njas

%I A081153
%S A081153 0,0,0,0,0,2,0,6,0,18,0,50,0,142,0,388,0,1114,0
%N A081153 Number of odd cycles in range [A014137(n-1)..A014138(n-1)] of permutation A057505/A057506, with no fixed points of either A057163 or A057164.
%Y A081153 a(n) = A081148(n)-A081155(n). Bisection: A081154.
%Y A081153 Adjacent sequences: A081150 A081151 A081152 this_sequence A081154 A081155 A081156
%Y A081153 Sequence in context: A076257 A046520 A019781 this_sequence A126869 A094233 A094659
%K A081153 nonn
%O A081153 0,6
%A A081153 Wouter Meeussen (wouter.meeussen(AT)pandora.be) and Antti Karttunen (Firstname.Surname(AT)iki.fi) Mar 10 2003

%I A126869
%S A126869 1,0,2,0,6,0,20,0,70,0,252,0,924,0,3432,0,12870,0,48620,0,184756,0,
%T A126869 705432,0,2704156,0,10400600,0,40116600,0,155117520,0,601080390,0,
%U A126869 2333606220,0,9075135300,0,35345263800,0,137846528820,0,538257874440,0
%N A126869 a(n)=Sum_{k, 0<=k<=n}binomial(n,floor(k/2))*(-1)^(n-k).
%C A126869 Hankel transform is 2^n . Successive binomial transforms are A002426, A000984, A026375, A081671, A098409, A098410.
%C A126869 Counts returning walks of length n on a 1-d integer lattice with step set {-1,+1}. - Andrew V. Sutherland (drew(AT)math.mit.edu), Feb 29 2008
%C A126869 Moment sequence of the trace of a random matrix in G=SO(2). If X=tr(A) is a random variable (A distributed with Haar measure on G), then a(n) = E[X^n]. - Andrew V. Sutherland (drew(AT)math.mit.edu), Feb 29 2008
%C A126869 Also the moment sequence of the trace of the kth power of a random matrix in USp(2)=SU(2), for all k > 2. - Andrew V. Sutherland (drew(AT)math.mit.edu), Feb 29 2008
%D A126869 Kiran S. Kedlaya and Andrew V. Sutherland, "Hyperelliptic curves, L-polynomials, and random matrices", preprint, 2008.
%H A126869 Kiran S. Kedlaya and Andrew V. Sutherland, <a href="http://arxiv.org/abs/0803.4462">Hyperelliptic curves, L-polynomials, and random matrices</a>.
%F A126869 a(2*n)=binomial(2*n,n)=A000984(n), a(2*n+1)=0 . a(n)=Sum_{k, 0<=k<=n}A107430(n,k)*(-1)^(n-k)=Sum_{k, 0<=k<=n}A061554(n,k)*(-1)^k.
%F A126869 a(n) = (1/Pi)*Integral_{t=0..Pi}cos^n(t)dt. - Andrew V. Sutherland (drew(AT)math.mit.edu), Feb 29 2008
%Y A126869 Cf. A107430, A061554.
%Y A126869 Cf. 126120.
%Y A126869 Adjacent sequences: A126866 A126867 A126868 this_sequence A126870 A126871 A126872
%Y A126869 Sequence in context: A046520 A019781 A081153 this_sequence A094233 A094659 A137437
%K A126869 nonn
%O A126869 0,3
%A A126869 Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 16 2007

%I A094233
%S A094233 1,0,2,0,6,0,20,0,70,2,252,22,924,156,3432,910,12870,4760,48622,23256,
%T A094233 184796,108528,705894,490314,2708204,2163150,10430500,9373652,40313160,
%U A094233 40060078,156305070,169345560,607812102,709645552,2369918628,2952780320
%N A094233 Number of closed walks of length n at a vertex of the cyclic graph on 9 nodes C_9.
%C A094233 In general a(n,m)=2^n/m*Sum_{k=0..m-1} Cos(2Pi*k/m)^n) counts closed walks of length n at a vertex of the cyclic graph on m nodes C_m.
%F A094233 a(n)=2^n/9*Sum_{k=0..8} Cos(2Pi*k/9)^n
%t A094233 f[n_] := FullSimplify[ TrigToExp[ 2^n/9 Sum[ Cos[2Pi*k/9]^n, {k, 0, 8}]]]; Table[ f[n], {n, 0, 40}] (from Robert G. Wilson v Jun 01 2004)
%Y A094233 Cf. A078008, A054877, A047849.
%Y A094233 Adjacent sequences: A094230 A094231 A094232 this_sequence A094234 A094235 A094236
%Y A094233 Sequence in context: A019781 A081153 A126869 this_sequence A094659 A137437 A021489
%K A094233 nonn
%O A094233 0,3
%A A094233 Herbert Kociemba (kociemba(AT)t-online.de), May 29 2004
%E A094233 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Jun 01 2004

%I A094659
%S A094659 1,0,2,0,6,0,20,2,70,18,252,110,924,572,3434,2730,12902,12376,48926,
%T A094659 54264,187036,232562,720062,980674,2789164,4086550,10861060,16878420,
%U A094659 42484682,69242082,166823430,282580872,657178982,1148548016,2595874468
%N A094659 Number of closed walks of length n at a vertex of the cyclic graph on 7 nodes C_7.
%C A094659 In general a(n,m)=2^n/m*Sum_{k=0..m-1} Cos(2Pi*k/m)^n) counts closed walks of length n at a vertex of the cyclic graph on m nodes C_m.
%F A094659 a(n)=2^n/7*Sum_{k=0..6} Cos(2Pi*k/7)^n); a(n)=7(a(n-2)-2a(n-4)+a(n-6))+2a(n-7); G.f.: (1-x-2x^2+x^3)/((2x-1)(-1-x+2x^2+x^3))
%t A094659 f[n_] := FullSimplify[ TrigToExp[ 2^n/7 Sum[Cos[2Pi*k/7]^n, {k, 0, 6}]]]; Table[ f[n], {n, 0, 36}] (from Robert G. Wilson v Jun 09 2004)
%Y A094659 Adjacent sequences: A094656 A094657 A094658 this_sequence A094660 A094661 A094662
%Y A094659 Sequence in context: A081153 A126869 A094233 this_sequence A137437 A021489 A092158
%K A094659 nonn
%O A094659 0,3
%A A094659 Herbert Kociemba (kociemba(AT)t-online.de), Jun 06 2004
%E A094659 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Jun 09 2004

%I A137437
%S A137437 1,0,0,0,2,0,6,0,24,0,120,40,0,720,420,0,5040,3948,0,40320,38304,2240,0,
%T A137437 362880,396576,50400
%V A137437 1,0,0,0,-2,0,6,0,-24,0,120,40,0,-720,-420,0,5040,3948,0,-40320,-38304,-2240,0,362880,
%W A137437 396576,50400
%N A137437 Triangular sequence from expansion coefficients of asymptotic Hermite Polynomial from Roman: p(x,t)= (1 + t)^(-x)*Exp[x*(t - t^2/2)].
%C A137437 Row sums are:A038205
%C A137437 {1, 0, 0, -2, 6, -24, 160, -1140, 8988, -80864, 809856};
%C A137437 These polynomials are unique in that they connect hyperbolic differential equations, derangements and Hermite orthogonal polynomials;
%C A137437 also the polynomials have a very slow rise in power with n.
%D A137437 Steve Roman , The Umbral Calculus, Dover Publications, New York (1984), page 130
%F A137437 p(x,t)= (1 + t)^(-x)*Exp[x*(t - t^2/2)]=Sum[q(x,n)*t^n/n!,{n,0,Infinity}]; out(n,m)=n!*Coefficients(q(x,n)).
%e A137437 {1},
%e A137437 {0},
%e A137437 {0},
%e A137437 {0, -2},
%e A137437 {0, 6},
%e A137437 {0, -24},
%e A137437 {0, 120, 40},
%e A137437 {0, -720, -420},
%e A137437 {0, 5040, 3948},
%e A137437 {0, -40320, -38304, -2240},
%e A137437 {0, 362880, 396576, 50400}
%t A137437 Clear[p, g, m, a]; p[t_] := (1 + t)^(-x)*Exp[x*(t - t^2/2)]; Table[ ExpandAll[n!SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]], {n, 0, 10}]; a = Table[n!* CoefficientList[SeriesCoefficient[ Series[p[t], {t, 0, 30}], n], x], {n, 0, 10}]; Flatten[{{1}, {0}, {0}, {0, -2}, {0, 6}, {0, -24}, {0, 120, 40}, {0, -720, -420}, {0, 5040, 3948}, {0, -40320, -38304, -2240}, {0, 362880, 396576, 50400}}]
%Y A137437 Cf. A038205, A137286.
%Y A137437 Adjacent sequences: A137434 A137435 A137436 this_sequence A137438 A137439 A137440
%Y A137437 Sequence in context: A126869 A094233 A094659 this_sequence A021489 A092158 A051831
%K A137437 nonn,uned,tabl,new
%O A137437 1,5
%A A137437 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 21 2008

%I A021489
%S A021489 0,0,2,0,6,1,8,5,5,6,7,0,1,0,3,0,9,2,7,8,3,5,0,5,1,5,4,6,3,9,1,7,5,
%T A021489 2,5,7,7,3,1,9,5,8,7,6,2,8,8,6,5,9,7,9,3,8,1,4,4,3,2,9,8,9,6,9,0,7,
%U A021489 2,1,6,4,9,4,8,4,5,3,6,0,8,2,4,7,4,2,2,6,8,0,4,1,2,3,7,1,1,3,4,0,2
%N A021489 Decimal expansion of 1/485.
%Y A021489 Adjacent sequences: A021486 A021487 A021488 this_sequence A021490 A021491 A021492
%Y A021489 Sequence in context: A094233 A094659 A137437 this_sequence A092158 A051831 A119883
%K A021489 nonn,cons
%O A021489 0,3
%A A021489 njas

%I A092158
%S A092158 2,0,6,1,9,4,4,4,1,6,7,6,0,0,6,7,7,4,1,3,5,4,1,2,3,2,9,5,8,3,3,5,7,7,8,
%T A092158 3,2,9,0,8,6,9,2,1,7,2,3,3,9,5,3,6,1,4,2,8,4,1,4,5,3,0,1,5,7,1,7,1,6,9,
%U A092158 1,8,0,7,4,4,1,8,0,6,2,1,5,4,5,8,5,3,7,1,8,4,4,6,7,4,1,9,3,7
%N A092158 Decimal expansion of (pi/e)^5.
%e A092158 2.0619444167600677
%Y A092158 Cf. A061382, A092035.
%Y A092158 Adjacent sequences: A092155 A092156 A092157 this_sequence A092159 A092160 A092161
%Y A092158 Sequence in context: A094659 A137437 A021489 this_sequence A051831 A119883 A020853
%K A092158 cons,nonn
%O A092158 1,1
%A A092158 Mohammad K. Azarian (azarian(AT)evansville.edu), Mar 31 2004

%I A051831
%S A051831 1,2,0,6,1,12,16,1,22,1,1,36,1,42,46,52,1,1,66,1,72,1,82,1,96,1,102,
%T A051831 106,1,112,126,1,136,1,1,1,156,162,166,172,1,1,1,192,196,1,1,222,226,1,
%U A051831 232,1,1,1,256,262,1,1,276,1,282,292,306,1,312,316,1,336,346,1,352,1
%N A051831 Fibonacci(Pn) mod Pn, where Pn is the n-th prime.
%C A051831 Terms are 1 when Pn == 1 or 4 mod 5, terms are Pn-1 when Pn == 2 or 3 mod 5.
%H A051831 R. Peele and P. Stanica, <a href="http://arXiv.org/abs/math.CO/0010186">Matrix powers of column-justified Pascal triangles and Fibonacci sequences</a>
%e A051831 P3=5, fibonacci(5)=5 == 0 mod 5.
%Y A051831 Cf. A000045, A003631, A045468, A051834, A051830.
%Y A051831 Adjacent sequences: A051828 A051829 A051830 this_sequence A051832 A051833 A051834
%Y A051831 Sequence in context: A137437 A021489 A092158 this_sequence A119883 A020853 A095832
%K A051831 nonn
%O A051831 1,2
%A A051831 Jud McCranie (j.mccranie(AT)comcast.net), Dec 11 1999

%I A119883
%S A119883 1,2,0,6,1,50,14,854,323,24930,11804,1111462,631621,70271890,46590634,
%T A119883 5980829430,4531805575,659311412930,562021682744,91385427666758,86555950096265,
%U A119883 15555589905976050,16206870089730374,3190048222084343446,3625755168948973771
%V A119883 1,2,0,-6,-1,50,14,-854,-323,24930,11804,-1111462,-631621,70271890,46590634,
%W A119883 -5980829430,-4531805575,659311412930,562021682744,-91385427666758,-86555950096265,
%X A119883 15555589905976050,16206870089730374,-3190048222084343446,-3625755168948973771
%N A119883 E.g.f. (1+2x+x^2/2)*sech(x).
%C A119883 Transform of binomial(2,n) under the matrix A119879.
%F A119883 a(n)=sum{k=0..n, A119879(n,k)*C(2,k)}
%Y A119883 Adjacent sequences: A119880 A119881 A119882 this_sequence A119884 A119885 A119886
%Y A119883 Sequence in context: A021489 A092158 A051831 this_sequence A020853 A095832 A036044
%K A119883 easy,sign
%O A119883 0,2
%A A119883 Paul Barry (pbarry(AT)wit.ie), May 26 2006

%I A020853
%S A020853 1,0,2,0,6,2,0,7,2,6,1,5,9,6,5,7,5,4,0,9,1,5,5,3,5,0,3,1,1,2,7,4,5,
%T A020853 4,7,4,6,6,5,2,4,7,8,1,1,6,9,4,0,2,7,9,2,2,0,1,8,0,2,8,8,5,6,9,6,8,
%U A020853 7,9,0,0,1,5,7,2,7,3,8,8,1,2,6,1,0,5,8,2,7,4,7,6,3,7,9,3,6,0,0,0,9
%N A020853 Decimal expansion of 1/sqrt(96).
%Y A020853 Adjacent sequences: A020850 A020851 A020852 this_sequence A020854 A020855 A020856
%Y A020853 Sequence in context: A092158 A051831 A119883 this_sequence A095832 A036044 A078991
%K A020853 nonn,cons
%O A020853 0,3
%A A020853 njas

%I A095832
%S A095832 0,2,0,6,2,0,12,6,2,0,20,12,6,2,0,30,20,12,6,2,0,42,30,20,12,6,2,0,56,
%T A095832 42,30,20,12,6,2,0,72,56,42,30,20,12,6,2,0,90,72,56,42,30,20,12,6,2,0,
%U A095832 110,90,72,56,42,30,20,12,6,2,0,132,110,90,72,56,42,30,20,12,6,2,0,156
%N A095832 Triangle read by rows: T(n,k) = (n-k+1)*(n-k), n>=1, 1<=k<=n.
%Y A095832 Adjacent sequences: A095829 A095830 A095831 this_sequence A095833 A095834 A095835
%Y A095832 Sequence in context: A051831 A119883 A020853 this_sequence A036044 A078991 A021833
%K A095832 easy,nonn,tabl
%O A095832 1,2
%A A095832 Herman Jamke (hermanjamke(AT)fastmail.fm), Jul 10 2004

%I A036044
%S A036044 1,0,2,0,6,2,4,0,14,6,10,2,12,4,8,0,30,14,22,6,26,10,18,2,28,12,20,4,
%T A036044 24,8,16,0,62,30,46,14,54,22,38,6,58,26,42,10,50,18,34,2,60,28,44,12,
%U A036044 52,20,36,4,56,24,40,8,48,16,32,0,126,62,94,30,110,46,78,14,118,54,86
%N A036044 BCR(n): write in binary, complement, reverse.
%H A036044 T. D. Noe, <a href="http://www.research.att.com/~njas/sequences/b036044.txt">Table of n, a(n) for n=0..1023</a>
%H A036044 R. K. Hoeflin, <a href="http://www.eskimo.com/~miyaguch/mega.html">Mega Test</a>
%F A036044 a(2n) = 2*A059894(n), a(2n+1) = a(2n) - 2^[log2(n)+1]. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Aug 21 2003
%e A036044 E.g. 4 -> 100 -> 011 -> 110 -> 6.
%t A036044 dtn[ L_ ] := Fold[ 2#1+#2&, 0, L ]; f[ n_ ] := dtn[ Reverse[ 1-IntegerDigits[ n, 2 ] ] ]; Table[ f[ n ], {n, 0, 100} ]
%Y A036044 Cf. A035928, A030101, A056539.
%Y A036044 Adjacent sequences: A036041 A036042 A036043 this_sequence A036045 A036046 A036047
%Y A036044 Sequence in context: A119883 A020853 A095832 this_sequence A078991 A021833 A049257
%K A036044 nonn,easy,base,nice
%O A036044 0,3
%A A036044 njas
%E A036044 More terms from Erich Friedman (erich.friedman(AT)stetson.edu).

%I A078991
%S A078991 1,0,1,0,1,2,0,6,2,6,0,36,24,12,24,0,240,240,240,72,120,0,1800,2400,
%T A078991 3600,2160,600,720,0,15120,25200,50400,45360,25200,5760,5040,0,141120,
%U A078991 282240,705600,846720,705600,322560,65520
%V A078991 1,0,1,0,1,-2,0,-6,-2,6,0,36,24,12,-24,0,-240,-240,-240,-72,120,0,1800,2400,3600,2160,
%W A078991 600,-720,0,-15120,-25200,-50400,-45360,-25200,-5760,5040,0,141120,282240,705600,
%X A078991 846720,705600,322560,65520
%N A078991 Coefficients of the polynomials in the numerator of the generating function x/(1-x-x^2) for the Fibonacci sequence and its successive derivatives starting with the highest power of x.
%F A078991 f(x)^(n), for n=0, 1, 2, 3, 4, . . ., where f(x)= x/(1-x-x^2)
%e A078991 The coefficients of the first 3 polynomials starting with the highest power of x are: 1,0; 1,0,1; -2,0,-6,-2; ...
%Y A078991 Adjacent sequences: A078988 A078989 A078990 this_sequence A078992 A078993 A078994
%Y A078991 Sequence in context: A020853 A095832 A036044 this_sequence A021833 A049257 A054877
%K A078991 sign,tabl
%O A078991 0,6
%A A078991 Mohammad K. Azarian (azarian(AT)evansville.edu), Jan 12 2003

%I A021833
%S A021833 0,0,1,2,0,6,2,7,2,6,1,7,6,1,1,5,8,0,2,1,7,1,2,9,0,7,1,1,7,0,0,8,4,
%T A021833 4,3,9,0,8,3,2,3,2,8,1,0,6,1,5,1,9,9,0,3,4,9,8,1,9,0,5,9,1,0,7,3,5,
%U A021833 8,2,6,2,9,6,7,4,3,0,6,3,9,3,2,4,4,8,7,3,3,4,1,3,7,5,1,5,0,7,8,4,0
%N A021833 Decimal expansion of 1/829.
%Y A021833 Adjacent sequences: A021830 A021831 A021832 this_sequence A021834 A021835 A021836
%Y A021833 Sequence in context: A095832 A036044 A078991 this_sequence A049257 A054877 A095834
%K A021833 nonn,cons
%O A021833 0,4
%A A021833 njas

%I A049257
%S A049257 2,0,6,2,13,12,2,0,9,2,7,6,2,4,3,2,1,0,18,18,18,18,1,18,2,18,18,2,18,
%T A049257 18,2,0,18,2,7,18,2,10,6,2,18,18,18,18,18,2,1,18,2,0,18,2,18,18,2,16,
%U A049257 15,2,13,12,2,4,9,8,7,6,5,4,3,2,1,0,18,18,18,18,18,18,2,18,18,2,13,18
%N A049257 Smallest nonnegative value taken on by 18x^2 - ny^2 for an infinite number of integer pairs (x, y).
%Y A049257 Adjacent sequences: A049254 A049255 A049256 this_sequence A049258 A049259 A049260
%Y A049257 Sequence in context: A036044 A078991 A021833 this_sequence A054877 A095834 A106828
%K A049257 nonn
%O A049257 1,1
%A A049257 David W. Wilson (davidwwilson(AT)comcast.net)

%I A054877
%S A054877 1,0,2,0,6,2,20,14,70,72,254,330,948,1430,3614,6008,13990,24786,
%T A054877 54740,101118,215766,409640,854702,1652090,3396916,6643782,13530350,
%U A054877 26667864,53971350,106914242,215492564,428292590,860941798
%N A054877 Closed walks of length n along the edges of a pentagon based at a vertex.
%C A054877 In general a(n,m)=2^n/m*Sum(k,0,m-1,Cos(2Pi*k/m)^n) counts closed walks of length n at a vertex of the cyclic graph on m nodes C_m. Here we have the case m=5. - Herbert Kociemba (kociemba(AT)t-online.de), May 31 2004
%F A054877 G.f.: -1/5*1/(2*x-1)-2/5*(2+x)/(x^2-x-1). a(n)=( 2^n + 2*(-1)^n*( F(n) + F(n-2) ) )/5, for n>1, where F(n) is the n-th Fibonacci number (cf. A000045)
%F A054877 a(n)=2^n/5*Sum(k, 0, 4, Cos(2Pi*k/5)^n) - Herbert Kociemba (kociemba(AT)t-online.de), May 31 2004
%F A054877 Recurrence: a(n)=5(a(n-2)-a(n-4)) + 2a(n-5) - Herbert Kociemba (kociemba(AT)t-online.de), Jun 04 2004
%Y A054877 {a(n)/2} for n>1 is A052964.
%Y A054877 Adjacent sequences: A054874 A054875 A054876 this_sequence A054878 A054879 A054880
%Y A054877 Sequence in context: A078991 A021833 A049257 this_sequence A095834 A106828 A055302
%K A054877 nonn,walk
%O A054877 0,3
%A A054877 Paolo Dominici (pl.dm(AT)libero.it), May 23 2000

%I A095834
%S A095834 0,2,0,6,3,0,12,8,4,0,20,15,10,5,0,30,24,18,12,6,0,42,35,28,21,14,7,0,
%T A095834 56,48,40,32,24,16,8,0,72,63,54,45,36,27,18,9,0,90,80,70,60,50,40,30,20,
%U A095834 10,0,110,99,88,77,66,55,44,33,22,11,0,132,120,108,96,84,72,60,48,36,24
%N A095834 Triangle read by rows: T(n,k) = (n-k)*n, n>=1, 1<=k<=n.
%Y A095834 Adjacent sequences: A095831 A095832 A095833 this_sequence A095835 A095836 A095837
%Y A095834 Sequence in context: A021833 A049257 A054877 this_sequence A106828 A055302 A055349
%K A095834 easy,nonn,tabl
%O A095834 1,2
%A A095834 Herman Jamke (hermanjamke(AT)fastmail.fm), Jul 10 2004

%I A106828
%S A106828 1,0,0,0,1,0,2,0,6,3,0,24,20,0,120,130,15,0,720,924,210,0,5040,7308,2380,105,0,40320,64224,
%T A106828 26432,2520,0,362880,623376,303660,44100,945,0,3628800,6636960,3678840,
%U A106828 705320,34650,0,39916800,76998240,47324376,11098780,866250,10395
%N A106828 Triangle T(n,k) read by rows: associated Stirling numbers of first kind (n >= 0, and, for n>=2, 0 <= k <= floor(n/2)).
%C A106828 Another version of the triangle in A008306, which see for formulae, references etc.
%D A106828 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 256.
%D A106828 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 75.
%e A106828 Rows 0 though 7 are:
%e A106828 1
%e A106828 0 0
%e A106828 0 1
%e A106828 0 2
%e A106828 0 6 3
%e A106828 0 24 20
%e A106828 0 120 130 15
%e A106828 0 720 924 210
%Y A106828 See A008306 for more information.
%Y A106828 Adjacent sequences: A106825 A106826 A106827 this_sequence A106829 A106830 A106831
%Y A106828 Sequence in context: A049257 A054877 A095834 this_sequence A055302 A055349 A136656
%K A106828 tabf,nonn,easy
%O A106828 0,7
%A A106828 njas, May 22 2005

%I A055302
%S A055302 1,2,0,6,3,0,24,36,4,0,120,360,140,5,0,720,3600,3000,450,6,0,5040,
%T A055302 37800,54600,18900,1302,7,0,40320,423360,940800,588000,101136,3528,8,0,
%U A055302 362880,5080320,16087680,15876000,5143824,486864,9144,9,0,3628800
%N A055302 Triangle of labeled rooted trees with n nodes and k leaves.
%H A055302 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/transforms.txt">Transforms</a>
%H A055302 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ro.html#rooted">Index entries for sequences related to rooted trees</a>
%F A055302 E.g.f. (relative to x) satisfies A(x, y)=xy+x*exp(A(x, y))-x. Divides by n and shifts up under exponential transform.
%F A055302 T(n, k) = (n!/k!)*Stirling2(n-1, n-k). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Jan 28 2004
%e A055302 1; 2,0; 6,3,0; 24,36,4,0; 120,360,140,5,0; ...
%Y A055302 Row sums give A000169. Columns 1 through 12: A000142, A055303-A055313. Cf. A055314.
%Y A055302 Adjacent sequences: A055299 A055300 A055301 this_sequence A055303 A055304 A055305
%Y A055302 Sequence in context: A054877 A095834 A106828 this_sequence A055349 A136656 A131595
%K A055302 nonn,tabl,eigen
%O A055302 1,2
%A A055302 Christian G. Bower (bowerc(AT)usa.net), May 11 2000

%I A055349
%S A055349 1,2,0,6,3,0,24,36,8,0,120,360,220,30,0,720,3600,4200,1500,144,0,5040,
%T A055349 37800,71400,47250,11508,840,0,40320,423360,1176000,1234800,545664,
%U A055349 98784,5760,0,362880,5080320,19474560,29635200,20469456,6618528
%N A055349 Triangle of labeled mobiles (circular rooted trees) with n nodes and k leaves.
%H A055349 <a href="http://www.research.att.com/~njas/sequences/Sindx_Mo.html#mobiles">Index entries for sequences related to mobiles</a>
%F A055349 E.g.f. satisfies A(x, y)=xy-x*log(1-A(x, y))-x.
%e A055349 1; 2,0; 6,3,0; 24,36,8,0; 120,360,220,30,0; ...
%Y A055349 Row sums give A038037. Columns 1 through 8: A000142, A055303, A055350-A055355.
%Y A055349 Adjacent sequences: A055346 A055347 A055348 this_sequence A055350 A055351 A055352
%Y A055349 Sequence in context: A095834 A106828 A055302 this_sequence A136656 A131595 A124228
%K A055349 nonn,tabl,eigen
%O A055349 1,2
%A A055349 Christian G. Bower (bowerc(AT)usa.net), May 15 2000

%I A136656
%S A136656 1,0,2,0,6,4,0,24,36,8,0,120,300,144,16,0,720,2640,2040,480,32,0,5040,
%T A136656 25200,27720,10320,1440,64,0,40320,262080,383040,199920,43680,4032,128,
%U A136656 0,362880,2963520,5503680,3764880,1142400,163968,10752,256,0,3628800
%V A136656 1,0,-2,0,6,4,0,-24,-36,-8,0,120,300,144,16,0,-720,-2640,-2040,-480,-32,0,5040,25200,
%W A136656 27720,10320,1440,64,0,-40320,-262080,-383040,-199920,-43680,-4032,-128,0,362880,
%X A136656 2963520,5503680,3764880,1142400,163968,10752,256,0,-3628800,-36288000,-82978560
%N A136656 Coefficients for rewriting generalized falling factorials into ordinary factorials.
%C A136656 Generalization of (signed) Lah number triangle A008297 (amended with a trivial row n=0 and a column k=0 in order to have a Sheffer triangle structure of the Jabotinsky type).
%C A136656 product(s*t-j,j=0..n-1) := fallfac(s*t,n) (falling factorial with n factors) is called generalized factorial of t of order n and scale parameter s in the Charalambides reference p.301 ch. 8.4.
%C A136656 The s-family of triangles L(s;n,k) (in the Charalambides reference called C(n,k;-s))
%C A136656 is defined for integer s by fallfac(-s*t,n) = ((-1)^n)*risefac(s*t,n) = sum(L(s;n,k)*fallfac(t,k),k=0..n), n>=0. risefac(x,n):=product(x+j,j=0..n-1) for the rising factorials.
%C A136656 For positive s the signless triangles |L(s;n,k)| = L(s;n,k)*(-1)^n satisfies risefac(s*t,n) = sum(|L(s;n,k)|*fallfac(t,k),k=0..n), n>=0.
%C A136656 For negative s see the combinatorial interpretation given in the Charalambides reference, Example 8.8, p. 313: Coupon collector's problem.
%D A136656 Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, ch. 8.4 p.301 ff. Table 8.3 (with row n=0 and column k=0 and s=-2).
%H A136656 W. Lang, <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A136656.text">First ten rows and more. </a>
%F A136656 Recurrence: a(n,k)= 0 if n<k; a(n,0)= 1 if n=0 else 0, and a(n,k)= (-2*k-(n-1))*a(n-1,k)-2*a(n-1,k-1). From the Charalambides reference Theorem 8.19, p.309, for s=-2.
%F A136656 E.g.f. column k: (1/(1+x)^2 - 1)^k/k!, k>=0. From the Charalambides reference Theorem 8.14, p.305 for s=-2.(hence a Sheffer triangle of Jabotinsky type).
%F A136656 a(n,k)=sum(((-1)^(k-r))*binomial(k,r)*fallfac(-2*r,n),r=0..k)/k!, n>=k>=0. From the Charalambides reference Theorem 8.15, p.306 for s=-2.
%F A136656 a(n,k)= sum(S1(n,r)*S2(r,k)*(-2)^r with the Stirling numbers S1(n,r)= A048993(n,r) and S2(r,k)= A048993(r,k). From the Charalambides reference Theorem 8.13, p.304 for s=-2.
%F A136656 a(n,k)= sum(M_3(n,k,q)*product(fallfac(-2,j)^e(n,m,q,j),j=1..n),q=1..p(n,k)) if n>=k>=1, else 0. Here p(n,k)=A008284(n,m), the number of k parts partitions of n, the M_3 partition numbers aregiven in A036040, and e(n,m,q,j) is the exponent of j in the q-th k parts partition of n. Rewritten eq. (8.50), Theorem 8.16, p. 307, from the Charalambides reference for s=-2.
%e A136656 [1];[0,-2];[0,6,4];[0,-24,-36,-8];[0,120,300,144,16];...
%e A136656 Recurrence: a(4,2) = -7*a(3,2)-2*a(3,1) = -7*(-36) -2*(-24) = 300.
%e A136656 a(4,2)=300 as sum over the M3 numbers A036040 for the 2 parts partitons of 4: 4*fallfac(-2,1)^1*fallfac(-2,3)^1 + 3*fallfac(-2,2)^2 = 4*(-2)*(-24)+3*6^2 = 300.
%Y A136656 Column sequences (unsigned) 2*A001710, 4*A136659, 8*A136660, 16*A136661 for k=1..4.
%Y A136656 Cf. A136657 without row n=0 and column k=0, divided by 2.
%Y A136656 Adjacent sequences: A136653 A136654 A136655 this_sequence A136657 A136658 A136659
%Y A136656 Sequence in context: A106828 A055302 A055349 this_sequence A131595 A124228 A115879
%K A136656 sign,easy,tabl
%O A136656 0,3
%A A136656 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Feb 22 2008

%I A131595
%S A131595 2,0,6,4,5,7,2,8,8,0,7,0,6,7,6,0,3,0,7,3,1,0,8,1,4,3,7,2,8,6,6
%N A131595 Decimal expansion of 3*(sqrt(25 + 10*sqrt(5))), the surface area of a regular dodecahedron with edges of unit length.
%C A131595 Surface area of a regular dodecahedron: A = 3*(sqrt(25 + 10*sqrt(5)))* a^2, where 'a' is the edge.
%e A131595 20.645728807067603...
%Y A131595 Cf. A102769.
%Y A131595 Adjacent sequences: A131592 A131593 A131594 this_sequence A131596 A131597 A131598
%Y A131595 Sequence in context: A055302 A055349 A136656 this_sequence A124228 A115879 A115880
%K A131595 cons,easy,nonn
%O A131595 2,1
%A A131595 Omar E. Pol (info(AT)polprimos.com), Aug 30 2007

%I A124228
%S A124228 0,1,0,2,0,6,4,10,8,20,16,32,32,58,60,96,104,162,180,260,296,416,480,
%T A124228 650,760,1012,1184,1540,1816,2330,2752,3476,4112,5142,6080,7522,8896,
%U A124228 10922,12900,15710,18536,22438,26432,31798,37400,44772,52560,62612
%N A124228 Number of partitions of n with odd crank.
%C A124228 For a partition p, let l(p) = largest part of p, w(p) = number of 1's in p, m(p) = number of parts of p larger than w(p). The crank of p is given by l(p) if w(p) = 0, otherwise m(p)-w(p).
%F A124228 a(n) = (A000041(n)-A124226(n))/2.
%p A124228 A000041 := proc(n) combinat[numbpart](n) ; end: A124226 := proc(n) local x,gf,i ; gf := 1; for i from 1 to n+1 do gf := taylor(gf*(1-x^i)/(1+x^i)^2,x=0,n+1) ; od ; coeftayl(2*x+gf,x=0,n) ; end: A124228 := proc(n) (A000041(n)-A124226(n))/2 ; end: for n from 0 to 60 do printf("%a, ",A124228(n)) ; od ; - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 18 2007
%Y A124228 Cf. A124226, A124227.
%Y A124228 Adjacent sequences: A124225 A124226 A124227 this_sequence A124229 A124230 A124231
%Y A124228 Sequence in context: A055349 A136656 A131595 this_sequence A115879 A115880 A078037
%K A124228 easy,nonn
%O A124228 0,4
%A A124228 Vladeta Jovovic (vladeta(AT)Eunet.yu), Oct 20 2006
%E A124228 More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 18 2007

%I A115879
%S A115879 0,0,2,0,6,4,12,3,6,12,30,8,42,24,4,6,72,12,90,24,10,60,132,5,30,84,18,
%T A115879 48,210,8,240,