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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Maintained by: N. J. A. Sloane (njas@research.att.com),
home page: www.research.att.com/~njas/
(start)
%I A021102
%S A021102 0,1,0,2,0,4,0,8,1,6,3,2,6,5,3,0,6,1,2,2,4,4,8,9,7,9,5,9,1,8,3,6,7,
%T A021102 3,4,6,9,3,8,7,7,5,5,1,0,2,0,4,0,8,1,6,3,2,6,5,3,0,6,1,2,2,4,4,8,9,
%U A021102 7,9,5,9,1,8,3,6,7,3,4,6,9,3,8,7,7,5,5,1,0,2,0,4,0,8,1,6,3,2,6,5,3
%N A021102 Decimal expansion of 1/98.
%Y A021102 Sequence in context: A131575 A077957 A077966 this_sequence A021053 A128983 A066493
%Y A021102 Adjacent sequences: A021099 A021100 A021101 this_sequence A021103 A021104 A021105
%K A021102 nonn,cons
%O A021102 0,4
%A A021102 njas
%I A021053
%S A021053 0,2,0,4,0,8,1,6,3,2,6,5,3,0,6,1,2,2,4,4,8,9,7,9,5,9,1,8,3,6,7,3,4,
%T A021053 6,9,3,8,7,7,5,5,1,0,2,0,4,0,8,1,6,3,2,6,5,3,0,6,1,2,2,4,4,8,9,7,9,
%U A021053 5,9,1,8,3,6,7,3,4,6,9,3,8,7,7,5,5,1,0,2,0,4,0,8,1,6,3,2,6,5,3,0,6
%N A021053 Decimal expansion of 1/49.
%Y A021053 Sequence in context: A077957 A077966 A021102 this_sequence A128983 A066493 A137449
%Y A021053 Adjacent sequences: A021050 A021051 A021052 this_sequence A021054 A021055 A021056
%K A021053 nonn,cons
%O A021053 0,2
%A A021053 njas
%I A128983
%S A128983 0,1,2,0,4,0,8,6,9,10,16,12,32,18,33,34,64,36,128,66,129,130,256,132,
%T A128983 257,258,134,260,512,264,1024,514,1025,1026,268,1028,2048,1032,2049,
%U A128983 2050,4096
%N A128983 Rightmost position of n in A089625, 0 if absent.
%C A128983 Numbers n have A000586(n) decompositions into sums of distinct primes and occur A000586(n) times in A089625. The sequence is the rightmost (largest) index (position) of n in A089625. It is an inverse of A089625 made unique in the sense that in the prime decomposition of n the one with the largest primes are chosen and converted to binary. The sequence therefore is a binary representation of a greedy decomposition of n into a sum of primes.
%F A128983 A089625(a(n))=n if n not equal to 1, 4 and 6.
%e A128983 Prime decompositions of n=25 are 1*11+1*7+1*5+0*3+1*2 (binary tagged 11101=29)
%e A128983 or 1*13+0*11+1*7+0*5+1*3+1*2 (binary 101011=43) or
%e A128983 1*13+0*11+1*7+1*5+0*3+0*2 (binary 101100=44) or 1*17+0*13+0*11+0*7+1*5+1*3+0*2
%e A128983 (binary 1000110=70) or 1*23+0*19+0*17+0*13+0*11+0*7+0*5+0*3+1*2 (binary 100000001
%e A128983 =257). Out of these indices 29, 43, 44, 70 and 257, the largest is chosen, a(25)=257.
%Y A128983 Cf. A089625, A000586.
%Y A128983 Sequence in context: A077966 A021102 A021053 this_sequence A066493 A137449 A056946
%Y A128983 Adjacent sequences: A128980 A128981 A128982 this_sequence A128984 A128985 A128986
%K A128983 nonn
%O A128983 1,3
%A A128983 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 30 2007
%I A066493
%S A066493 1,2,0,4,0,9,0,24,0,34,0,46,0,30,0,282,0,99,0,154,0,189,0,263,0,367,0,
%T A066493 429,0,590,0,738,0,217,0,1183,0,3302,0,2191,0,1879,0,1831,0,7970,0,
%U A066493 3077,0,3427
%N A066493 a(n) = least k such that f(k) = n, where f is the prime gaps function given by f(m) = p(m+1)-p(m) and p(m) denotes the m-th prime, if k exists; 0 otherwise.
%C A066493 Obviously, a(n) = 0 for every odd n except 1. From the list, it appears that a(n) is nonzero for every even n; is this true in general? That is, for each even n, are there primes which differ by n?
%e A066493 a(6) = 9 since k = 9 is the smallest k making p(k+1)-p(k) = 6. a(3) = 0 since no two primes differ by 3.
%t A066493 f[n_] := Prime[n + 1] - Prime[n]; g[n_] := Min[Select[Range[1, 10^4], f[ # ] == n &]]; Table[g[i], {i, 1, 50}]
%Y A066493 Sequence in context: A021102 A021053 A128983 this_sequence A137449 A056946 A111757
%Y A066493 Adjacent sequences: A066490 A066491 A066492 this_sequence A066494 A066495 A066496
%K A066493 nonn
%O A066493 1,2
%A A066493 Joseph L. Pe (joseph_l_pe(AT)hotmail.com), Jan 03 2002
%I A137449
%S A137449 1,1,1,2,0,4,0,12,0,40,12,0,48,0,176,0,120,0,800,0,608,120,0,720,0,5280,
%T A137449 0,1856,0,1680,0,16800,0,25536,0,5248,1680,0,13440,0,147840,0,103936,0,
%U A137449 14080,0,30240,0,403200,0,919296,0,377856,0,36352,30240,0,302400,0
%V A137449 1,1,1,-2,0,-4,0,-12,0,-40,12,0,48,0,-176,0,120,0,800,0,-608,-120,0,-720,0,5280,0,
%W A137449 -1856,0,-1680,0,-16800,0,25536,0,-5248,1680,0,13440,0,-147840,0,103936,0,-14080,0,
%X A137449 30240,0,403200,0,-919296,0,377856,0,-36352,-30240,0,-302400,0,4435200,0,-4677120,0
%N A137449 A triangular sequence based on concepts of operations on existing sequences: in this case the H(x,n) ( A060821) traditional Hermite is differentiated twice : p(x,n)=-x^2*H''(x,n)+H(x,n).
%C A137449 Row sums are:
%C A137449 {1, 2, -6, -52, -116, 312, 2584, 1808, -42864, -144352, 601504};
%C A137449 As an operator algebra like an Energy Hamiltonian:
%C A137449 e(n)*H(x,n)=p(x,n)/x^2
%C A137449 The relative energy of the row sums goes up much faster than in the Chebyshev
%C A137449 of the first kind.
%F A137449 p(x,n)=-x^2*H''(x,n)+H(x,n)
%e A137449 {1},
%e A137449 {1, 1},
%e A137449 {-2, 0, -4},
%e A137449 {0, -12, 0, -40},
%e A137449 {12, 0, 48, 0, -176},
%e A137449 {0, 120,0, 800, 0, -608},
%e A137449 {-120, 0, -720, 0, 5280, 0, -1856},
%e A137449 {0, -1680, 0, -16800, 0, 25536, 0, -5248},
%e A137449 {1680, 0, 13440, 0, -147840, 0, 103936, 0, -14080},
%e A137449 {0, 30240, 0, 403200, 0, -919296, 0, 377856, 0, -36352},
%e A137449 {-30240, 0, -302400, 0, 4435200, 0, -4677120,0, 1267200, 0, -91136}
%t A137449 Clear[p, x, a] p[x, 0] = 1; p[x, 1] = x + 1; p[x_, n_] := p[x, n] = -x^2*D[HermiteH[n, x], {x, 2}] + HermiteH[n, x]; Table[Expand[p[x, n]], {n, 0, 10}]; a = Table[CoefficientList[p[x, n], x], {n, 0, 10}]; Flatten[a]
%Y A137449 Sequence in context: A021053 A128983 A066493 this_sequence A056946 A111757 A022896
%Y A137449 Adjacent sequences: A137446 A137447 A137448 this_sequence A137450 A137451 A137452
%K A137449 nonn,tabl,uned
%O A137449 1,4
%A A137449 Roger L. Bagula and Gary Adamson (rlbagulatftn(AT)yahoo.com), Apr 18 2008
%I A056946
%S A056946 1,2,0,4,0,12,0,32728,196560,262154,0,3734484,16773120,18345916,0,103029576,
%T A056946 398034000,376741188,0,1334312620,4629381120,3937755904,0,10792611336,34417656000,
%U A056946 26962933262,0,62783799320,187489935360,138065611740,0,287105506144,814879774800
%V A056946 1,2,0,-4,0,12,0,32728,196560,262154,0,3734484,16773120,18345916,0,103029576,
%W A056946 398034000,376741188,0,1334312620,4629381120,3937755904,0,10792611336,34417656000,
%X A056946 26962933262,0,62783799320,187489935360,138065611740,0,287105506144,814879774800
%N A056946 Coefficients of J(0)*theta_3(z) where J(0) is sequence A056945.
%C A056946 Note a(4n)=A008408(n) gives theta series of Leech lattice. a(4n+2)=0. a(3)<0, apparently the only negative term. What are a(4n+1), a(4n+3)?
%D A056946 Eichler and Zagier, The Theory of Jacobi Forms, Birkhauser, 1985.
%F A056946 (E_8*E_{4, 1}-60*E_{12, 1})*theta_3(z)
%Y A056946 Cf. A008408, A056945.
%Y A056946 Sequence in context: A128983 A066493 A137449 this_sequence A111757 A022896 A100225
%Y A056946 Adjacent sequences: A056943 A056944 A056945 this_sequence A056947 A056948 A056949
%K A056946 sign
%O A056946 0,2
%A A056946 Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jul 17 2000
%I A111757
%S A111757 0,1,0,1,0,2,0,4,0,13,0,48,0,238,0,1325,0,8297,0,54519
%N A111757 Number of bipartite 2-connected outerplanar graphs on n unlabeled nodes.
%C A111757 Also the number of bipartite (unlabeled) dissections of a polygon.
%H A111757 M. Bordirsky, E. Fusy, M. Kang and S. Vigerske, Enumeration of Unlabeled Outerplanar Graphs, 2005
%H A111757 S. Vigerske, Asymptotic enumeration of unlabeled outerplanar graphs, Diploma thesis, Humboldt University Berlin, 2005
%H A111757 S. Vigerske, Homepage
%F A111757 Generating function and cycle index sum known, see Vigerske.
%Y A111757 Cf. A001004.
%Y A111757 Sequence in context: A066493 A137449 A056946 this_sequence A022896 A100225 A007420
%Y A111757 Adjacent sequences: A111754 A111755 A111756 this_sequence A111758 A111759 A111760
%K A111757 nonn
%O A111757 1,6
%A A111757 Stefan Vigerske Nov 21 2005
%I A022896
%S A022896 1,0,0,0,0,0,2,0,4,0,14,0,40,0,125,0,394,0,1294,0,4356
%N A022896 Number of solutions to c(1)p(1)+...+c(n)p(n) = 2, where c(i) = +-1 for i>1, c(1) = 1, p(i) = primes.
%Y A022896 Sequence in context: A137449 A056946 A111757 this_sequence A100225 A007420 A019219
%Y A022896 Adjacent sequences: A022893 A022894 A022895 this_sequence A022897 A022898 A022899
%K A022896 nonn
%O A022896 1,7
%A A022896 Clark Kimberling (ck6(AT)evansville.edu)
%I A100225
%S A100225 1,1,2,0,4,0,16,0,80,0,448,0,2688,0,16896,0,109824,0,732160,0,4978688,0,34398208,
%T A100225 0,240787456,0,1704034304,0,12171673600,0,87636049920,0,635361361920,0,
%U A100225 4634400522240,0,33985603829760,0,250420238745600,0,1853109766717440
%V A100225 1,1,2,0,-4,0,16,0,-80,0,448,0,-2688,0,16896,0,-109824,0,732160,0,-4978688,0,34398208,
%W A100225 0,-240787456,0,1704034304,0,-12171673600,0,87636049920,0,-635361361920,0,
%X A100225 4634400522240,0,-33985603829760,0,250420238745600,0,-1853109766717440
%N A100225 G.f. A(x) satisfies: 3^n - 1 = Sum_{k=0..n} [x^k]A(x)^n, and also satisfies: (3+z)^n - (1+z)^n + z^n = Sum_{k=0..n} [x^k](A(x)+z*x)^n for all z, where [x^k]A(x)^n denotes the coefficient of x^k in A(x)^n.
%C A100225 More generally, if g.f. A(x) satisfies: m^n-b^n = Sum_{k=0..n} [x^k]A(x)^n, then A(x) also satisfies: (m+z)^n - (b+z)^n + z^n = Sum_{k=0..n} [x^k](A(x)+z*x)^n for all z, and A(x)=(1+(m-1)*x+sqrt(1+2*(m-2*b-1)*x+(m^2-2*m+4*b+1)*x^2))/2.
%F A100225 G.f.: (1+2*x+sqrt(1+8*x^2))/2. G.f.: A(x) = x/(series_reversion[x*(1-x)/(1-2*x-x^2)]). a(n) = -8*(n-3)*a(n-2)/n for n>2, with a(0)=1, a(1)=1, a(2)=2. a(2*n) = 2^n*(-1)^(n-1)*A000108(n-1), a(2*n+1)=0, for n>=1, where A000108=Catalan.
%e A100225 From the table of powers of A(x) (A100226), we see that
%e A100225 3^n-1 = Sum of coefficients [x^0] through [x^n] in A(x)^n:
%e A100225 A^1=[1,1],2,0,-4,0,16,0,-80,...
%e A100225 A^2=[1,2,5],4,-4,-8,16,32,-80,...
%e A100225 A^3=[1,3,9,13],6,-12,-4,48,0,...
%e A100225 A^4=[1,4,14,28,33],8,-24,16,80,...
%e A100225 A^5=[1,5,20,50,85,81],10,-40,60,..
%e A100225 A^6=[1,6,27,80,171,246,197],12,-60,...
%e A100225 the main diagonal of which is A100227 = [1,5,13,33,81,197,477,...],
%e A100225 where Sum_{n>=1} A100227(n)/n*x^n = log((1-x)/(1-2*x-x^2).
%o A100225 (PARI) {a(n)=if(n==0,1,(3^n-1-sum(k=0,n,polcoeff(sum(j=0,min(k,n-1),a(j)*x^j)^n+x*O(x^k),k)))/n)} (PARI) {a(n)=if(n==0,1,if(n==1,1,if(n==2,2,-8*(n-3)*a(n-2)/n)))} (PARI) {a(n)=polcoeff((1+2*x+sqrt(1+8*x^2+x^2*O(x^n)))/2,n)}
%Y A100225 Cf. A100226, A100227, A000108, A025225, A100223, A100228.
%Y A100225 Sequence in context: A056946 A111757 A022896 this_sequence A007420 A019219 A019139
%Y A100225 Adjacent sequences: A100222 A100223 A100224 this_sequence A100226 A100227 A100228
%K A100225 sign
%O A100225 0,3
%A A100225 Paul D. Hanna (pauldhanna(AT)juno.com), Nov 28 2004
%I A007420 M0030
%S A007420 0,0,1,2,0,4,0,16,16,32,64,64,256,0,768,512,2048,3072,4096,
%T A007420 12288,4096,40960,16384,114688,131072,262144,589824,393216,
%U A007420 2097152,262144,6291456,5242880,15728640,27262976,29360128
%V A007420 0,0,1,2,0,-4,0,16,16,-32,-64,64,256,0,-768,-512,2048,3072,-4096,
%W A007420 -12288,4096,40960,16384,-114688,-131072,262144,589824,-393216,
%X A007420 -2097152,-262144,6291456,5242880,-15728640,-27262976,29360128
%N A007420 Berstel sequence: a(n+1)=2a(n)-4a(n-1)+4a(n-2).
%C A007420 a(n) = 0 only for n = 0,1,4,6,13 and 52. [Beukers] - Robert G. Wilson v (rgwv(AT)rgwv.com), Jul 05 2000
%D A007420 F. Beukers, The zero-multiplicity of ternary recurrences, Composito Math. 77 (1991), 165-177.
%D A007420 G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; p. 28.
%D A007420 Myerson, G. and van der Poorten, A. J., Some problems concerning recurrence sequences, Amer. Math. Monthly 102 (1995), no. 8, 698-705.
%D A007420 J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 193.
%F A007420 G.f.: x^2/(1-2*x+4*x^2-4*x^3).
%p A007420 A007420 := proc(n) options remember; if n <=1 then 0 elif n=2 then 1 else 2*A007420(n-1)-4*A007420(n-2)+4*A007420(n-3); fi; end;
%Y A007420 Cf. A035302.
%Y A007420 Sequence in context: A111757 A022896 A100225 this_sequence A019219 A019139 A022904
%Y A007420 Adjacent sequences: A007417 A007418 A007419 this_sequence A007421 A007422 A007423
%K A007420 sign,easy,nice
%O A007420 0,4
%A A007420 njas
%I A019219
%S A019219 0,2,0,4,0,18,18,100,188,568,1062,2922,6026,16486,38740,105022,
%T A019219 257446
%N A019219 Cycle class sequence c(n) (the number of true cycles of length n in which a certain node is included) for zeolite RSN = RUB-17 K4Na12 [ Zn8Si28O72 ] . 18 H2O.
%D A019219 G. Thimm and W. E. Klee, Zeolite cycle sequences, Zeolites, 19, pp. 422-424, 1997.
%H A019219 G. Thimm, Cycle sequences of crystal structures
%Y A019219 Sequence in context: A022896 A100225 A007420 this_sequence A019139 A022904 A019215
%Y A019219 Adjacent sequences: A019216 A019217 A019218 this_sequence A019220 A019221 A019222
%K A019219 nonn
%O A019219 3,2
%A A019219 Georg Thimm (mgeorg(AT)ntu.edu.sg)
%I A019139
%S A019139 0,2,0,4,0,18,18,100,190,566,1044,2906,6052,16554,38714,105048,
%T A019139 256738
%N A019139 Cycle class sequence c(n) (the number of true cycles of length n in which a certain node is included) for zeolite LOV = Lovdarite K4Na12 [ Be8Si28O72 ] . 18 H2O.
%D A019139 G. Thimm and W. E. Klee, Zeolite cycle sequences, Zeolites, 19, pp. 422-424, 1997.
%H A019139 G. Thimm, Cycle sequences of crystal structures
%Y A019139 Sequence in context: A100225 A007420 A019219 this_sequence A022904 A019215 A112081
%Y A019139 Adjacent sequences: A019136 A019137 A019138 this_sequence A019140 A019141 A019142
%K A019139 nonn
%O A019139 3,2
%A A019139 Georg Thimm (mgeorg(AT)ntu.edu.sg)
%I A022904
%S A022904 0,0,0,0,0,0,1,0,2,0,4,0,31,0,106,0,272,0,1135
%N A022904 Number of solutions to c(1)p(4)+...+c(n)p(n+3) = 1, where c(i) = +-1 for i>1, c(1) = 1, p(i) = primes.
%Y A022904 Sequence in context: A007420 A019219 A019139 this_sequence A019215 A112081 A056859
%Y A022904 Adjacent sequences: A022901 A022902 A022903 this_sequence A022905 A022906 A022907
%K A022904 nonn
%O A022904 1,9
%A A022904 Clark Kimberling (ck6(AT)evansville.edu)
%I A019215
%S A019215 0,2,0,4,0,40,0,150,4,610,96,2798,864,13218,7192,64934,52332,328222
%N A019215 Cycle class sequence c(n) (the number of true cycles of length n in which a certain node is included) for zeolite RON = Roggianite Ca16[ Be8Al16Si32O104(OH)16 ] . 19 H2O.
%D A019215 G. Thimm and W. E. Klee, Zeolite cycle sequences, Zeolites, 19, pp. 422-424, 1997.
%H A019215 G. Thimm, Cycle sequences of crystal structures
%Y A019215 Sequence in context: A019219 A019139 A022904 this_sequence A112081 A056859 A090888
%Y A019215 Adjacent sequences: A019212 A019213 A019214 this_sequence A019216 A019217 A019218
%K A019215 nonn
%O A019215 3,2
%A A019215 Georg Thimm (mgeorg(AT)ntu.edu.sg)
%I A112081
%S A112081 0,1,1,0,2,0,4,1,0,0,0,6,3,0,6,0,1,1,0,3,0,9,3,0,0,0,1,3,0,2,0,1,1,0,9,
%T A112081 0,1,1,0,12,0,3,1,0,0,0,3,4,0,2,0,1,1,0,3,0,16,3,0,0,0,1,3,0,5,0,1,1,0,
%U A112081 14,0,1,1,0,2,0,19,1,0,0,0,3,3,0,2,0,1,1,0
%V A112081 0,1,-1,0,2,0,4,-1,0,0,0,6,-3,0,6,0,1,-1,0,-3,0,9,-3,0,0,0,1,-3,0,-2,0,1,-1,0,9,0,1,-1,
%W A112081 0,12,0,3,-1,0,0,0,3,-4,0,2,0,1,-1,0,-3,0,16,-3,0,0,0,1,-3,0,-5,0,1,-1,0,14,0,1,-1,0,2,
%X A112081 0,19,-1,0,0,0,3,-3,0,2,0,1,-1,0
%N A112081 A112080(n)/2.
%Y A112081 Sequence in context: A019139 A022904 A019215 this_sequence A056859 A090888 A020781
%Y A112081 Adjacent sequences: A112078 A112079 A112080 this_sequence A112082 A112083 A112084
%K A112081 sign
%O A112081 1,5
%A A112081 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Aug 27 2005
%I A056859
%S A056859 1,2,0,4,1,0,8,7,0,0,16,32,4,0,0,32,121,49,1,0,0,64,411,360,42,0,0,0,
%T A056859 128,1304,2062,624,22,0,0,0,256,3949,10163,6042,730,7,0,0,0,512,11567,
%U A056859 45298,45810,12170,617,1,0,0,0,1024,33056,187941,296017,141822,18325
%N A056859 Triangle of number of falls in set partitions of n.
%C A056859 Number of falls s_i > s_{i+1} in a set partition {s_1, ..., s_n} of {1, ..., n}, where s_i is the subset containing i, s(1) = 1, and s(i) <= 1 + max of previous s(j)'s.
%C A056859 The maximum number of falls is in a set partition like 1,2,1,3,2,1,... - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jun 08 2006
%D A056859 W. C. Yang, Conjectures on some sequences involving set partitions and Bell numbers, preprint, 2000.
%e A056859 For example {1, 2, 1, 2, 2, 3} is a set partition of {1, 2, 3, 4, 5, 6} and has 1 fall, at i = 2.
%e A056859 1; 2,0; 4,1,0; 8,7,0,0; 16,32,4,0,0; 32,121,49,1,0,0; ...
%Y A056859 Cf. Bell numbers A000110.
%Y A056859 Cf. A056857-A056863.
%Y A056859 Sequence in context: A022904 A019215 A112081 this_sequence A090888 A020781 A007432
%Y A056859 Adjacent sequences: A056856 A056857 A056858 this_sequence A056860 A056861 A056862
%K A056859 easy,nonn,tabf
%O A056859 1,2
%A A056859 Winston C. Yang (winston(AT)cs.wisc.edu), Aug 31 2000
%E A056859 Corrected and extended by Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jun 08 2006
%I A090888
%S A090888 1,2,0,4,1,1,8,5,3,1,16,19,9,4,2,32,65,27,14,7,3,64,211,81,46,23,11,5,
%T A090888 128,665,243,146,73,37,18,8,256,2059,729,454,227,119,60,29,13,512,6305,
%U A090888 2187,1394,697,373,192,97,47,21,1024,19171,6561,4246,2123,1151,600,311
%N A090888 Matrix defined by a(n,k) = 3^n(Fibonacci(k)) - 2^n(Fibonacci(k-2)), read by antidiagonals.
%C A090888 a(0,k) = A000045(k-1); a(1,k) = A000032(k); a(2,k) = A000285(k+1).
%C A090888 a(n,1) = a(n-1,1) + a(n-1,3) for n > 0; a(n,1) = A001047(n) = 2^(2n) - A083324(n); a(n,2) = A000244(n) = 2^(2n) - A005061(n); a(n,3) = 2a(n-1,4) for n > 0; a(n,3) = A027649(n); a(n,4) = A083313(n+1); a(n,5) = A084171(n+1).
%C A090888 Sum[a(n-k,k), {k,0,n}] = A098703(n+1).
%C A090888 Let R, S, and T be binary relations on the power set P(A) of a set A having n = |A| elements such that for every element x, y of P(A), xRy if x is a subset of y or y is a subset of x, xSy if x is a subset of y, and xTy if x is a proper subset of y. Then a(n,3) = |R|, a(n,2) = |S|, and a(n,1) = |T|. Note that a binary relation W on P(A) can be defined also such that for every element x, y of P(A) xWy if x is a proper subset of y and there are no z in P(A) such that x is a proper subset of z and z is a proper subset of y. A090802(n,1) = |W|. Also, a(n,0) = |P(A)|.
%H A090888 Eric Weisstein, Fibonacci Number
%H A090888 Eric Weisstein, Lucas Number
%F A090888 a(n, k) = 3^n(Fibonacci(k)) - 2^n(Fibonacci(k-2)).
%F A090888 a(n, 0) = 2^n, a(n, 1) = 3^n - 2^n, a(n, k) = a(n, k-1) + a(n, k-2) for k > 1.
%F A090888 a(0, k) = Fibonacci(k-1), a(1, k) = Lucas(k), a(n, k) = 5a(n-1, k) - 6a(n-2, k) for n > 1.
%F A090888 O.g.f. (by rows) = (-2^n + (2^(n+1) - 3^n)x)/(-1+x+x^2). - Ross La Haye (rlahaye(AT)new.rr.com), Mar 30 2006
%F A090888 a(n,1) - a(n,0) = A003063(n+1). - Ross La Haye (rlahaye(AT)new.rr.com), Jun 22 2007
%F A090888 Binomial transform (by columns) of A118654. - Ross La Haye (rlahaye(AT)new.rr.com), Jun 22 2007
%e A090888 {1}; {2,0}; {4,1,1}; {8,5,3,1}; {16,19,9,4,2}; {32,65,27,14,7,3};
%e A090888 {64,211,81,46,23,11,5}; {128,665,243,146,73,37,18,8}
%e A090888 a(5,3) = 454 because Fibonacci(3) = 2, Fibonacci(1) = 1, and (2 * 3^5) - (1 * 2^5) = 454.
%Y A090888 Sequence in context: A019215 A112081 A056859 this_sequence A020781 A007432 A079124
%Y A090888 Adjacent sequences: A090885 A090886 A090887 this_sequence A090889 A090890 A090891
%K A090888 nonn,tabl
%O A090888 0,2
%A A090888 Ross La Haye (rlahaye(AT)new.rr.com), Feb 12 2004; revised Sep 24 2004, Sep 10 2005.
%E A090888 More terms from Ray Chandler (rayjchandler(AT)sbcglobal.net), Oct 27 2004
%I A020781
%S A020781 2,0,4,1,2,4,1,4,5,2,3,1,9,3,1,5,0,8,1,8,3,1,0,7,0,0,6,2,2,5,4,9,0,
%T A020781 9,4,9,3,3,0,4,9,5,6,2,3,3,8,8,0,5,5,8,4,4,0,3,6,0,5,7,7,1,3,9,3,7,
%U A020781 5,8,0,0,3,1,4,5,4,7,7,6,2,5,2,2,1,1,6,5,4,9,5,2,7,5,8,7,2,0,0,1,9
%N A020781 Decimal expansion of 1/sqrt(24).
%Y A020781 Sequence in context: A112081 A056859 A090888 this_sequence A007432 A079124 A056737
%Y A020781 Adjacent sequences: A020778 A020779 A020780 this_sequence A020782 A020783 A020784
%K A020781 nonn,cons
%O A020781 0,1
%A A020781 njas
%I A007432 M0031
%S A007432 1,1,0,1,2,0,4,1,3,2,8,0,10,4,0,2,14,3,16,2,0,8,20,0,13,
%T A007432 10,8,4,26,0,28,4,0,14,8,3,34,16,0,2,38,0,40,8,6,20,44,
%U A007432 0,31,13,0,10,50,8,16,4,0,26,56,0,58,28,12,8,20,0,64,14
%V A007432 1,-1,0,1,2,0,4,1,3,-2,8,0,10,-4,0,2,14,-3,16,2,0,-8,20,0,13,
%W A007432 -10,8,4,26,0,28,4,0,-14,8,3,34,-16,0,2,38,0,40,8,6,-20,44,
%X A007432 0,31,-13,0,10,50,-8,16,4,0,-26,56,0,58,-28,12,8,20,0,64,14
%N A007432 Moebius transform applied thrice to natural numbers.
%H A007432 T. D. Noe, Table of n, a(n) for n=1..1000
%H A007432 M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210.
%H A007432 N. J. A. Sloane, Transforms
%F A007432 Multiplicative with a(p^e) = sum (-1)^k (3 choose k)p^(e-k)[e>=k], k=0..3
%F A007432 Dirichlet g.f.: zeta(x-1)/zeta^3(x)
%Y A007432 Sequence in context: A056859 A090888 A020781 this_sequence A079124 A056737 A008797
%Y A007432 Adjacent sequences: A007429 A007430 A007431 this_sequence A007433 A007434 A007435
%K A007432 sign,easy,nice,mult
%O A007432 1,5
%A A007432 njas
%I A079124
%S A079124 1,0,1,0,2,0,4,1,5,1,11,0,17,4,13,13,37,2,53,13,51,35,103,10,135,78,167,
%T A079124 89,255,4,339,253,378,306,542,121,759,558,872,498,1259,121,1609,1180,
%U A079124 1677,1665,2589,808,3250,1969,3844,3325,5119,1850,6268,4758,7546,7070
%N A079124 Number of ways to partition n into distinct positive integers <= phi(n), where phi is Euler's totient function (A000010).
%F A079124 a(n) = b(0, n), b(m, n) = 1 + sum(b(i, j): m= square root(n) and the greatest divisor of n that is <= square root(n).
%F A056737 a(n) = Min{t - d | 0 < d <= t <= n and d*t=n}. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 25 2002
%e A056737 a(8) = 2 because 8 = 2*(2+2) and 8 = k*(k+1) or 8 = k^2 have no solutions for k = a positive integer.
%t A056737 A033676[n_] := If[EvenQ[DivisorSigma[0, n]], Divisors[n][[DivisorSigma[0, n]/2]], Sqrt[n]] A033677[n_] := If[EvenQ[DivisorSigma[0, n]], Divisors[n][[DivisorSigma[0, n]/2+1]], Sqrt[n]] Table[A033677[n] - A033676[n], {n, 1, 128}] (Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 27 2004)
%Y A056737 Sequence in context: A020781 A007432 A079124 this_sequence A008797 A109468 A081880
%Y A056737 Adjacent sequences: A056734 A056735 A056736 this_sequence A056738 A056739 A056740
%K A056737 nonn
%O A056737 1,3
%A A056737 Leroy Quet (qq-quet(AT)mindspring.com), Aug 26 2000
%I A008797
%S A008797 1,0,2,0,4,1,6,2,9,4,12,6,16,9,20,12,25,16,30,20,36,25,
%T A008797 42,30,49,36,56,42,64,49,72,56,81,64,90,72,100,81,110,90,
%U A008797 121,100,132,110,144,121,156,132,169,144,182,156,196,169
%N A008797 Molien series for 3-dimensional group [2,n ]+ = 22n.
%H A008797 Index entries for Molien series
%p A008797 (1+x^5)/(1-x^2)^2/(1-x^4)
%Y A008797 Sequence in context: A007432 A079124 A056737 this_sequence A109468 A081880 A037035
%Y A008797 Adjacent sequences: A008794 A008795 A008796 this_sequence A008798 A008799 A008800
%K A008797 nonn
%O A008797 0,3
%A A008797 njas
%I A109468
%S A109468 1,2,0,4,2,0,0,0,8,32,0,8,0,0,0,0,0,64,0,1968,508,0,0,0,16
%N A109468 a(n) is the number of permutations of (1,2,3,...,n) written in binary such that no adjacent elements share a common 1-bit.
%C A109468 In other words, if b(m) and b(m+1) are adjacent elements written in binary, then (b(m) AND b(m+1)) = 0 for 1 <= m <= n-1. (If a logical AND is applied to each pair of adjacent terms, the result is zero.)
%C A109468 Let 2^k be the largest power of 2 <= n. Note that element 2^k-1 can be adjacent only to 2^k. So 2^k-1 must be at the beginning or the end of the permutation while 2^k must be next to 2^k-1. The elements 2^k-1-2^i (i=1,...,k-1) can be adjacent only to 2^i, 2^k, and 2^k+2^i implying that n must be >=2^k+2^(k-3) to yield a nonzero number of permutations.
%Y A109468 Sequence in context: A079124 A056737 A008797 this_sequence A081880 A037035 A112824
%Y A109468 Adjacent sequences: A109465 A109466 A109467 this_sequence A109469 A109470 A109471
%K A109468 nonn
%O A109468 1,2
%A A109468 njas, based on a suggestion from Leroy Quet, Aug 21 2005
%E A109468 More terms from Max Alekseyev (maxal(AT)cs.ucsd.edu), Aug 28 2005
%I A081880
%S A081880 0,2,0,4,2,0,6,8,10,16,42,10,16,42,12,14,32,170,4816,3865642,2490531345360,16,42,18,20,66,22,80,1066,
%T A081880 189392,5978221610,5956522269711832016,5913359591595499145281505571167104042,
%U A081880 5827970276585748074286667660065476529979312208145367609757859954142122960,24,26
%N A081880 Triangle read by rows: n-th row gives trajectory of 2n under the map x->(x^2-4)/6, stopping when the next term would be negative or nonintegral.
%H A081880 Pierre Abbat, The 64-100 Sequences
%e A081880 8 -> (64-4)/6 = 10 -> (100-4)/6 = 16 -> (256-4)/6 = 42 -> (42^2-4)/6 nonintegral, so stop; thus row 4 is (8, 10, 16, 42).
%Y A081880 Sequence in context: A056737 A008797 A109468 this_sequence A037035 A112824 A001100
%Y A081880 Adjacent sequences: A081877 A081878 A081879 this_sequence A081881 A081882 A081883
%K A081880 nonn,tabf
%O A081880 0,2
%A A081880 Pierre Abbat, phma(AT)webjockey.net, Apr 12, 2003
%I A037035
%S A037035 0,0,0,2,0,4,2,2,0,8,6,4,2,16,26,2,0,28,2,20,6,16,14,8,42,34,14,28,2,
%T A037035 10,2,10,14,16,24,52,30,8,6,22,14,26,14,28,6,58,14,4,20,68,54,20,20,4,
%U A037035 158,2,80,8,68,130,32,14,134,28,12,130,8,2,32,28,24,10,14,28,36,32,14
%N A037035 Least k such that 2^n+1+k is a prime.
%e A037035 a(5)=4 because 2^5+1+4=37 that is a prime.
%Y A037035 Cf. A016014.
%Y A037035 A013597(n) - 1.
%Y A037035 Sequence in context: A008797 A109468 A081880 this_sequence A112824 A001100 A136265
%Y A037035 Adjacent sequences: A037032 A037033 A037034 this_sequence A037036 A037037 A037038
%K A037035 nonn
%O A037035 0,4
%A A037035 Felice Russo (felice.russo(AT)katamail.com)
%E A037035 More terms from Erich Friedman (erich.friedman(AT)stetson.edu)
%I A112824
%S A112824 0,0,0,2,0,4,2,2,4,8,6,10,6,6,10,14,12,12,14,14,10,20,14,16,18,16,16,24,
%T A112824 22,28,20,24,24,26,26,34,26,32,30,38,36,40,36,36,28,42,36,18,44,38,40,
%U A112824 50,42,40,50,48,40,54,52,48,42,46,42,56,56,64,48,60,64,68,66,66,48,60
%N A112824 Consider the Goldbach conjecture that every even number 2n=p+p' with p<=p'. Consider all such Goldbach partitions; a(n) is the difference between the largest p and the smallest p. Call this difference the Goldbach gap.
%C A112824 The gap is always even.
%F A112824 A112823 - A020481.
%t A112824 f[n_] := Block[{p = 2, q = n/2}, While[ !PrimeQ[p] || !PrimeQ[n - p], p++ ]; While[ !PrimeQ[q] || !PrimeQ[n - q], q-- ]; q - p]; Table[ f[n], {n, 4, 150, 2}]
%Y A112824 Cf. A020481.
%Y A112824 Sequence in context: A109468 A081880 A037035 this_sequence A001100 A136265 A066910
%Y A112824 Adjacent sequences: A112821 A112822 A112823 this_sequence A112825 A112826 A112827
%K A112824 nonn
%O A112824 2,4
%A A112824 Robert G. Wilson v (rgwv(AT)rgwv.com), Sep 05 2005
%I A001100
%S A001100 1,0,2,0,4,2,2,10,10,2,14,40,48,16,2,90,230,256,120,22,2,646,1580,1670,
%T A001100 888,226,28,2,5242,12434,12846,7198,2198,366,34,2,47622,110320,112820,64968,
%U A001100 22120,4448,540,40,2,479306,1090270,1108612,650644,236968,54304,7900,748
%N A001100 Triangle read by rows: T(n,k) = number of permutations of length n with exactly k rising or falling successions, for n >= 1, 0 <= k <= n-1.
%C A001100 Number of permutations of 12...n such that exactly k of the following occur: 12, 23, ..., (n-1)n, 21, 32, ..., n(n-1).
%D A001100 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263.
%D A001100 J. Riordan, A recurrence for permutations without rising or falling successions. Ann. Math. Statist. 36 (1965), 708-710.
%F A001100 Let S_{n, k} = number of permutations of 12...n with exactly k rising or falling successions. Let S[n](t) = Sum_{k >= 0} S_{n, k}*t^k. Then S[0] = 1; S[1] = 1; S[2] = 2*t; S[3] = 4*t+2*t^2; for n >= 4, S[n] = (n+1-t)*S[n-1] - (1-t)*(n-2+3*t)*S[n-2] - (1-t)^2*(n-5+t)*S[n-3] + (1-t)^3*(n-3)*S[n-4].
%e A001100 1; 0,2; 0,4,2; 2,10,10,2; 14,40,48,16,2; ...
%Y A001100 Diagonals give A002464, A086852, A086853, A086854, A086955.
%Y A001100 Triangle in A086856 multiplied by 2. Cf. A010028.
%Y A001100 Sequence in context: A081880 A037035 A112824 this_sequence A136265 A066910 A094405
%Y A001100 Adjacent sequences: A001097 A001098 A001099 this_sequence A001101 A001102 A001103
%K A001100 tabl,nonn
%O A001100 1,3
%A A001100 njas, Aug 19 2003
%I A136265
%S A136265 1,1,2,0,4,2,3,2,12,4,0,16,6,32,8,5,2,60,16,80,16,0,36,10,192,40,192,32,
%T A136265 7,2,168,36,560,96,448,64,0,64,14,640,112,1536,224,1024,128,9,2,360,64,
%U A136265 2160,320,4032,512,2304,256,0,100,18,1600,240,6720,864,10240,1152,5120
%V A136265 1,-1,2,0,-4,2,3,-2,-12,4,0,16,-6,-32,8,-5,2,60,-16,-80,16,0,-36,10,192,-40,-192,32,7,
%W A136265 -2,-168,36,560,-96,-448,64,0,64,-14,-640,112,1536,-224,-1024,128,-9,2,360,-64,-2160,
%X A136265 320,4032,-512,-2304,256,0,-100,18,1600,-240,-6720,864,10240,-1152,-5120,512,11,-2
%N A136265 Integral form of A053120 :Triangle of coefficients of Integral form Chebyshev's T(n, x) polynomials (powers of x in increasing order); Much improved version by use of the integro-differential recursive form over a previous attempt.
%C A136265 Row sums are:
%C A136265 Join[{1}, Table[Apply[Plus, CoefficientList[2*x*P[x, n] - Q[x, n], x]], {n,
%C A136265 0, 10}]];
%C A136265 {1, 1, -2, -7, -14, -23, -34, -47, -62, -79, -98, -119}
%C A136265 Integration of the doubled functions is not orthogonal:
%C A136265 Table[Table[Integrate[Sqrt[1/(1 - x^2)]*(2*x*P[x, n] - Q[x, n])*(2*x*P[x, m] -
%C A136265 Q[x, m]), {x, -1, 1}], {n, 0, 10}], {m, 0, 10}]
%D A136265 Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986,Pages 42-50
%F A136265 P(x, n) = 2*x*P(x, n - 1) - P(x, n - 2); Q(x, n) := D[P[x, n + 1], x]=dp(x,n)/dx Output Integral form: IP(x,n)=2*x*p(x,n)-Q(x,n)
%e A136265 {1},
%e A136265 {-1, 2},
%e A136265 {0, -4, 2},
%e A136265 {3, -2, -12, 4},
%e A136265 {0, 16, -6, -32, 8},
%e A136265 {-5,2, 60, -16, -80, 16},
%e A136265 {0, -36, 10, 192, -40, -192, 32},
%e A136265 {7, -2, -168, 36, 560, -96, -448, 64},
%e A136265 {0, 64, -14, -640, 112, 1536, -224, -1024, 128},
%e A136265 {-9, 2, 360, -64, -2160, 320, 4032, -512, -2304, 256},
%e A136265 {0, -100, 18, 1600, -240, -6720, 864, 10240, -1152, -5120, 512},
%e A136265 {11, -2, -660, 100,6160, -800, -19712, 2240, 25344, -2560, -11264, 1024}
%t A136265 P[x, 0] = 1; P[x, 1] = x; P[x_, n_] := P[x, n] = 2*x*P[x, n - 1] - P[x, n - 2]; Q[x_, n_] := D[P[x, n + 1], x]; Table[ExpandAll[2*x*P[x, n] - Q[x, n]], {n, 0, 10}]; a = Join[{{1}}, Table[CoefficientList[2*x*P[x, n] - Q[x, n], x], {n, 0, 10}]]; Join[{1}, Table[Apply[Plus, CoefficientList[2*x*P[x, n] - Q[x, n], x]], {n, 0, 10}]]; Flatten[a]
%Y A136265 Cf. A053120.
%Y A136265 Sequence in context: A037035 A112824 A001100 this_sequence A066910 A094405 A028609
%Y A136265 Adjacent sequences: A136262 A136263 A136264 this_sequence A136266 A136267 A136268
%K A136265 nonn,uned,tabl
%O A136265 1,3
%A A136265 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Mar 18 2008
%I A066910
%S A066910 1,0,1,2,0,4,2,3,5,0,8,4,6,10,4,5,7,11,1,17,11,18,10,15,1,21,11,16,26,
%T A066910 17,27,16,24,7,5,1,29,13,17,25,1,33,15,20,30,5,45,33,7,2,42,22,32,52,
%U A066910 38,8,2,47,23,32,50,25,35,55,31,46,10,3,57,29,41,65,41,64,36,53,11,2
%N A066910 a(1) = 1; a(n+1) = (sum{k=1 to n} a(k) ) (mod n).
%C A066910 Steven Taschuk and Phil Carmody posted to sci.math (http://www.mathforum.com/epigone/sci.math/sazhazhi ) that a(k) = 97 for k >= 398.
%e A066910 a(7) = (1 + 0 + 1 + 2 + 0 + 4) (mod 6) = 8 (mod 6) = 2.
%Y A066910 Sequence in context: A112824 A001100 A136265 this_sequence A094405 A028609 A107490
%Y A066910 Adjacent sequences: A066907 A066908 A066909 this_sequence A066911 A066912 A066913
%K A066910 easy,nonn
%O A066910 1,4
%A A066910 Leroy Quet (qq-quet(AT)mindspring.com), Jan 22 2002
%I A094405
%S A094405 1,1,2,0,4,2,3,5,0,8,4,6,10,4,5,7,11,1,17,11,18,10,15,1,21,11,16,26,17,
%T A094405 27,16,24,7,5,1,29,13,17,25,1,33,15,20,30,5,45,33,7,2,42,22,32,52,38,8,
%U A094405 2,47,23,32,50,25,35,55,31,46,10,3,57,29,41,65,41,64,36,53,11,2,62,26
%N A094405 a(1) = 1; a(n) = (sum of previous terms) mod n.
%C A094405 Theorem. For all values of n>=397, a(n)=97. Proof. Let s(n) denote Sum[a(i), i=1..n-1]. Calculation shows that s(397)=38606=397*97+97. Thus a(397)=397*97+97 mod 397=97. Then s(398)=s(397)+97=398*97+97, giving a(398)=97. A simple inductive argument shows that a(397+k)=97 for all integers k>=0. - John W. Layman (layman(AT)math.vt.edu), Jun 07 2004
%C A094405 Conjecture: For any seed a(1) the sequence "a(n) = (sum of previous terms) mod n" ends with repeating constant. This is true for a(1) = 1,...,941. - Zak Seidov (zakseidov(AT)yahoo.com), Feb 24 2006
%e A094405 a(4) = 0 because the previous terms 1, 1, 2 sum to 4, and 4 mod 4 is 0. a(5) = 4 because the previous terms 1, 1, 2, 0 sum to 4 and 4 mod 5 is 4.
%p A094405 L := [1]; s := 1; p := 2; while (nops(L) < 90) do; if 1>0 then; t := s mod p; L := [op(L),t]; s := s+t; p := p+1; fi; od; L;
%Y A094405 Sequence in context: A001100 A136265 A066910 this_sequence A028609 A107490 A079534
%Y A094405 Adjacent sequences: A094402 A094403 A094404 this_sequence A094406 A094407 A094408
%K A094405 nonn
%O A094405 1,3
%A A094405 Chuck Seggelin (seqfan(AT)plastereddragon.com), Jun 03 2004
%I A028609
%S A028609 1,2,0,4,2,4,0,0,0,6,0,2,4,0,0,8,2,0,0,0,4,0,0,4,0,6,0,8,0,0,0,4,0,4,0,
%T A028609 0,6,4,0,0,0,0,0,0,2,12,0,4,4,2,0,0,0,4,0,4,0,0,0,4,8,0,0,0,2,0,0,4,0,8,
%U A028609 0,4,0,0,0,12,0,0,0,0,4,10,0,0,0,0,0,0,0,4,0,0,4,8,0,0,0,4,0,6,6,0,0
%N A028609 Expansion of (theta_3(z)*theta_3(11z)+theta_2(z)*theta_2(11z)).
%C A028609 Theta series of lattice with Gram matrix [2, 1; 1, 6].
%C A028609 Number of integer solutions (x,y) to x^2+xy+3y^2=n. - Michael Somos, Sep 20 2004
%D A028609 H. McKean and V. Moll, Elliptic Curves, Cambridge University Press, 1997, page 202. MR1471703 (98g:14032)
%H A028609 John Cannon, Table of n, a(n) for n = 0..5000
%F A028609 Moebius transform is period 11 sequence [ 2, -2, 2, 2, 2, -2, -2, -2, 2, -2, 0, ...]. - Michael Somos Jan 29 2007
%F A028609 a(n)=2*b(n) and b(n) is multiplicative with b(11^e) = 1, b(p^e) = (1+(-1)^e)/2 if p == 2,6,7,8,10 (mod 11), b(p^e) = e+1 if p == 1,3,4,5,9 (mod 11) . - Michael Somos Jan 29 2007
%F A028609 G.f.: 1 +2 Sum_{k>0} kronecker(-11,n)*x^n/(1-x^n) . - Michael Somos Jan 29 2007
%F A028609 G.f. is Fourier series of a weight 1 level 11 modular form. f(-1/ (11 t)) = sqrt(11) (t/i) f(t) where q = exp(2 pi i t) . - Michael Somos Jun 05 2007
%e A028609 1 + 2*q^2 + 4*q^6 + 2*q^8 + 4*q^10 + 6*q^18 + 2*q^22 + 4*q^24 + 8*q^30 + 2*q^32 + 4*q^40 + 4*q^46 + 6*q^50 + 8*q^54 + 4*q^62 + 4*q^66 + 6*q^72 + 4*q^74 + ...
%o A028609 (PARI) a(n)=local(t); if(n<1,n==0, 2*issquare(n) +2*sum(y=1,sqrtint(n*4\11), 2*issquare(t=4*n-11*y^2)-(t==0))) /* Michael Somos, Sep 20 2004 */
%o A028609 (PARI) a(n)=if(n<1, n==0, qfrep([2,1;1,6],n,1)[n]*2) /* Michael Somos Jun 05 2005 */
%o A028609 (PARI) a(n)=if(n<1, n==0, direuler(p=2,n,1/(1-X)/(1-kronecker(-11,p)*X))[n]*2) /* Michael Somos Jun 05 2005 */
%o A028609 (PARI) {a(n)=if(n<1, n==0, 2*sumdiv(n, d, kronecker(-11, d)))} /* Michael Somos Jan 29 2007 */
%Y A028609 a(n)=2*A035179(n) if n>0.
%Y A028609 Sequence in context: A136265 A066910 A094405 this_sequence A107490 A079534 A097042
%Y A028609 Adjacent sequences: A028606 A028607 A028608 this_sequence A028610 A028611 A028612
%K A028609 nonn
%O A028609 0,2
%A A028609 njas
%I A107490
%S A107490 1,2,0,4,2,4,0,0,0,6,0,4,8
%N A107490 Coefficients of a certain theta series.
%C A107490 See reference for details.
%D A107490 W. R. Parry, A negative result on the representation of modular forms by theta series, J. Reine Angew. Math., 310 (1979), 151-170.
%Y A107490 Sequence in context: A066910 A094405 A028609 this_sequence A079534 A097042 A097927
%Y A107490 Adjacent sequences: A107487 A107488 A107489 this_sequence A107491 A107492 A107493
%K A107490 nonn
%O A107490 0,2
%A A107490 njas, May 28 2005
%I A079534
%S A079534 0,1,1,2,0,4,2,4,2,8,2,10,4,6,6,13,3,15,5,9,7,19,5,17,9,15,9,25,4,26,12,
%T A079534 16,12,20,8,32,14,20,12,36,8,38,15,19,17,41,11,37,15,27,19,47,13,35,19,
%U A079534 31,23,52,10,54,24,30,26,42,14,60,26,38,18,64,18,66,30,33,29,53,17,71,25
%N A079534 phi(n) - ceiling( (log 2 / 2) * (n / log n) ).
%C A079534 It is known that a(n) >= 0.
%D A079534 D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, p. 9.
%Y A079534 Cf. A000010, A079530, A079531, A079532, A079533.
%Y A079534 Sequence in context: A094405 A028609 A107490 this_sequence A097042 A097927 A097945
%Y A079534 Adjacent sequences: A079531 A079532 A079533 this_sequence A079535 A079536 A079537
%K A079534 nonn
%O A079534 2,4
%A A079534 njas, Jan 23 2003
%I A097042
%S A097042 1,2,0,4,2,4,4,8,8,10,12,16,20,24,28,36,42,48,60,72,84,100,116,136,160,
%T A097042 186,216,252,292,336,388,448,512,588,672,768,878,1000,1136,1292,1464,1656,
%U A097042 1876,2120,2388,2696,3032,3408,3832,4298,4816,5396,6036,6744,7532,8404
%N A097042 G.f. = (G.f. for A096661)/(1 + 2 Sum_{m>=1} (-1)^m*q^(m^2)).
%C A097042 a(0) = 1; for n>0, a(n) = 2*A026832(n) (i.e. essentially Fine's numbers L(n) multiplied by 2).
%D A097042 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 56, Eq. (26.28).
%Y A097042 Cf. A096661, A026832.
%Y A097042 Sequence in context: A028609 A107490 A079534 this_sequence A097927 A097945 A083218
%Y A097042 Adjacent sequences: A097039 A097040 A097041 this_sequence A097043 A097044 A097045
%K A097042 nonn
%O A097042 0,2
%A A097042 njas, Sep 15 2004
%I A097927
%S A097927 1,1,2,0,4,2,6,0,0,4,10,0,12,6,8,0,16,0,18,0,12,10,22,0,0,12,0,0,28,8,30,0
%N A097927 Duplicate of A097945.
%Y A097927 Sequence in context: A107490 A079534 A097042 this_sequence A097945 A083218 A139716
%Y A097927 Adjacent sequences: A097924 A097925 A097926 this_sequence A097928 A097929 A097930
%K A097927 dead
%O A097927 1,3
%I A097945
%S A097945 1,1,2,0,4,2,6,0,0,4,10,0,12,6,8,0,16,0,18,0,12,10,22,0,0,12,0,0,28,8,
%T A097945 30,0,20,16,24,0,36,18,24,0,40,12,42,0,0,22,46,0,0,0,32,0,52,0,40,0,36,
%U A097945 28,58,0,60,30,0,0,48,20,66,0,44,24,70,0,72,36,0,0,60,24,78,0,0,40,82,0
%V A097945 1,-1,-2,0,-4,2,-6,0,0,4,-10,0,-12,6,8,0,-16,0,-18,0,12,10,-22,0,0,12,0,0,-28,-8,-30,0,
%W A097945 20,16,24,0,-36,18,24,0,-40,-12,-42,0,0,22,-46,0,0,0,32,0,-52,0,40,0,36,28,-58,0,-60,
%X A097945 30,0,0,48,-20,-66,0,44,-24,-70,0,-72,36,0,0,60,-24,-78,0,0,40,-82,0
%N A097945 a(n) = mu(n)*phi(n) where mu(n) is the Mobius function (A008683) and phi(n) is the Euler totient function (A000010).
%C A097945 Also, a(n) = mu(n)*uphi(n) where mu(n) is the Mobius function (A008683) and uphi(n) is the unitary totient function (A047994), since phi(n) = uphi(n) when n is square-free, while mu(n) = 0 when n is not square-free. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), May 14 2006
%C A097945 Conjecture: Sum_n=1..inf mu(n)/phi(n) = Sum_n=1..inf a(n)/phi(n)^2 = 0 It is true that Sum_n=1..inf mu(n)/phi(n)^s = 0 at least for s > 1 since: phi(2)=1, phi is multiplicative, so for n's that are square-free, the phi(n) values can be partitioned in pairs where phi(m)=phi(2m) and mu(m) = -mu(2m). So Sum_i=1..n mu(i)/phi(i)^s < Sum j=[n/2]..n 1/phi(j)^s which approaches 0 as n increases since 1) n^(1-e) < phi(n) < n for any e > 0 and n > N(e) and 2) Sum_i..n 1/n^s converges for s > 1. Conjecture: Sum_n=1..inf mu(n)/phi(n)^z = 0 for Re(z) > 1
%C A097945 Multiplicative with a(p^1) = 1-p, a(p^e) = 0, e > 1. Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu) May 24, 2005.
%C A097945 Row sums of triangle A143153 = a signed version of the sequence such that parity = (-) iff A008683(n) = (+); 0 or (+): (1, 1, 2, 0, 4, -2, 6, 0, 0, -4, 10, 0, 12, -6, 0, 0, 0,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 27 2008
%H A097945 Euler's totient function at Wikipedia.org
%t A097945 Table[ MoebiusMu[n]EulerPhi[n], {n, 85}] (from Robert G. Wilson v Sep 06 2004)
%Y A097945 Cf. A000010, A008683, A047994.
%Y A097945 Sequence in context: A079534 A097042 A097927 this_sequence A083218 A139716 A068773
%Y A097945 Cf. A143153.
%Y A097945 Adjacent sequences: A097942 A097943 A097944 this_sequence A097946 A097947 A097948
%K A097945 sign,mult
%O A097945 1,3
%A A097945 Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Sep 04 2004
%E A097945 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Sep 06 2004
%E A097945 Edited by njas, May 20 2006
%I A083218
%S A083218 1,2,0,4,2,6,1,0,0,10,2,12,1,2,0,16,2,18,1,0,0,22,2,0,1,2,0,28,2,30,1,0,
%T A083218 0,4,2,36,1,2,0,40,2,42,1,0,0,46,2,0,1,2,0,52,2,0,1,0,0,58,2,60,1,2,0,4,
%U A083218 2,66,1,0,0,70,2,72,1,2,0,4,2,78,1,0,0,82,2,0,1,2,0,88,2,2,1,0,0,4,2,96
%N A083218 a(n) = n mod (spf(n+1)+1), where spf(n) is the smallest prime factor of n (A020639).
%C A083218 a(n) = n iff n+1 is prime: a(A006093(k))=A006093(k).
%Y A083218 Cf. A057237.
%Y A083218 Sequence in context: A097042 A097927 A097945 this_sequence A139716 A068773 A133168
%Y A083218 Adjacent sequences: A083215 A083216 A083217 this_sequence A083219 A083220 A083221
%K A083218 nonn
%O A083218 1,2
%A A083218 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 22 2003
%I A139716
%S A139716 0,1,2,0,4,2,6,4,0,6,10,3,12,10,6,0,16,9,18,4
%N A139716 If k = the largest divisor of n that is <= sqrt(n), then a(n) = n - k^2.
%C A139716 a(p) = p-1 for all primes p. a(n^2) = 0 for all positive integers n.
%Y A139716 Cf. A139717, A033676.
%Y A139716 Sequence in context: A097927 A097945 A083218 this_sequence A068773 A133168 A078909
%Y A139716 Adjacent sequences: A139713 A139714 A139715 this_sequence A139717 A139718 A139719
%K A139716 more,nonn
%O A139716 1,3
%A A139716 Leroy Quet (qq-quet(AT)mindspring.com), May 01 2008
%I A068773
%S A068773 1,0,2,0,4,2,8,4,10,6,16,12,24,18,26,18,34,28,46,38,50,40,62,54,74,62,
%T A068773 80,68,96,88,118,102,122,106,130,118,154,136,160,144,184,172,214,194,
%U A068773 218,196,242,226,268,248,280,256,308,290,330,306,342,314,372,356
%N A068773 Alternating sum eulerphi(1)-eulerphi(2)+eulerphi(3)-eulerphi(4)+...+((-1)^(n+1))*eulerphi(n).
%F A068773 a(n) = sum((-1)^(k+1)*eulerphi(k), k=1..n)
%e A068773 a(3) = eulerphi(1) - eulerphi(2) + eulerphi(3) = 1 - 1 + 2 = 2.
%o A068773 (PARI) a068773(m)=local(s,n); s=0; for(n=1,m, if(n%2==0,s=s-eulerphi(n),s=s+eulerphi(n)); print1(s,","))
%Y A068773 Cf. A000010, A067929.
%Y A068773 Sequence in context: A097945 A083218 A139716 this_sequence A133168 A078909 A067458
%Y A068773 Adjacent sequences: A068770 A068771 A068772 this_sequence A068774 A068775 A068776
%K A068773 easy,nonn
%O A068773 1,3
%A A068773 Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Feb 28 2002
%I A133168
%S A133168 0,0,0,1,0,2,0,4,2,13,7,78,106,839,2368,16228,69292,471625,2629867
%N A133168 Minimal transposition classes of latin trades of size n.
%D A133168 I. M. Wanless, A computer enumeration of small latin trades, Australas. J. Combin. 39, (2007) 247-258.
%Y A133168 Cf. A007083.
%Y A133168 Sequence in context: A083218 A139716 A068773 this_sequence A078909 A067458 A088330
%Y A133168 Adjacent sequences: A133165 A133166 A133167 this_sequence A133169 A133170 A133171
%K A133168 nonn
%O A133168 1,6
%A A133168 Ian Wanless (ian.wanless(AT)sci.monash.edu.au), Oct 11 2007
%I A078909
%S A078909 0,2,0,4,3,2,0,6,0,5,0,4,5,2,3,8,5,2,0,7,0,2,0,6,6,7,0,4,7,5,0,10,0,7,
%T A078909 3,4,7,2,5,9,9,2,0,4,3,2,0,8,0,8,5,9,9,2,3,6,0,9,0,7,11,2,0,12,8,2,0,9,
%U A078909 0,5,0,6,11,9,6,4,0,7,0,11,0,11,0,4,8,2,7,6,13,5,5,4,0,2,3,10,13,2,0
%N A078909 Let r+i*s be the sum, with multiplicity, of the first-quadrant Gaussian primes dividing n; sequence gives s values.
%C A078909 A Gaussian integer z = x+iy is in the first quadrant if x > 0, y >= 0. Just one of the 4 associates z, -z, i*z, -i*z is in the first quadrant.
%C A078909 The sequence is fully additive.
%H A078909 Michael Somos, PARI program for finding prime decomposition of Gaussian integers
%H A078909 Index entries for Gaussian integers and primes
%e A078909 5 factors into the product of the primes 1+2*i, 1-2*i, but the first-quadrant associate of 1-2*i is i*(1-2*i) = 2+i, so r+i*s = 1+2*i + 2+i = 3+3*i. Therefore a(5) = 3.
%Y A078909 Cf. A078458, A078908, A078910, A078911, A080088, A080089.
%Y A078909 Sequence in context: A139716 A068773 A133168 this_sequence A067458 A088330 A128263
%Y A078909 Adjacent sequences: A078906 A078907 A078908 this_sequence A078910 A078911 A078912
%K A078909 nonn,easy
%O A078909 1,2
%A A078909 njas, Jan 11 2003
%E A078909 More terms and further information from Vladeta Jovovic (vladeta(AT)Eunet.yu), Jan 27 2003
%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 Sequence in context: A068773 A133168 A078909 this_sequence A088330 A128263 A140254
%Y A067458 Adjacent sequences: A067455 A067456 A067457 this_sequence A067459 A067460 A067461
%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 Sequence in context: A133168 A078909 A067458 this_sequence A128263 A140254 A095202
%Y A088330 Adjacent sequences: A088327 A088328 A088329 this_sequence A088331 A088332 A088333
%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, Modular Forms Database.
%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 Sequence in context: A078909 A067458 A088330 this_sequence A140254 A095202 A093443
%Y A128263 Adjacent sequences: A128260 A128261 A128262 this_sequence A128264 A128265 A128266
%K A128263 sign,mult
%O A128263 1,5
%A A128263 Michael Somos, Feb 21 2007
%I A140254
%S A140254 1,1,2,0,4,3,6,0,0,5,10,0,12,7,6,0,16,0,18,0,8,11,22,0,0,13,0,0,28,7,30,
%T A140254 0,12,17,10,0,36,19,14,0,40,9,42,0,0,23,46,0,0,0,18,0,52,0,14,0,20,29,
%U A140254 58,0,60,31,0,0,16,13,66,0,24,11,70,0,72,37,0,0,16
%V A140254 1,1,2,0,4,-3,6,0,0,-5,10,0,12,-7,-6,0,16,0,18,0,-8,-11,22,0,0,-13,0,0,28,7,30,0,-12,
%W A140254 -17,-10,0,36,-19,-14,0,40,9,42,0,0,-23,46,0,0,0,-18,0,52,0,-14,0,-20,-29,58,0,60,-31,
%X A140254 0,0,-16,13,66,0,-24,11,70,0,72,-37,0,0,-16
%N A140254 Mobius transform of A014963.
%C A140254 Conjectures relating to the Mobius sequence A008683:
%C A140254 If mu(n) = 0, a(n) = 0.
%C A140254 If mu(n) = 1, (n>1), a(n) = a negative term.
%C A140254 If mu(n) = -1, a(n) = a positive term.
%C A140254 So except for the first term and zero divided by zero we would have mu(n) = -a(n)/abs(a(n)).
%C A140254 Examples: mu(4) = 0, a(4) = 0; mu(6) = 1, a(6) = (-3); mu(7) = (-1), a(7) = 6.
%H A140254 Physics Forums discussion, Moebius function.
%H A140254 Eric. W. Weisstein, Mertens Conjecture.
%F A140254 A054525 as an infinite lower triangular matrix * A014963 as a vector.
%e A140254 a(5) = -3 = (1, -1, -1, 0, 0, 1) dot (1, 2, 3, 2, 5, 1) = (1 - 2 - 3 + 0 + 0 + 1), where (1, -1, -1, 0, 0, 1) = row 5 of triangle A054525 and (1, 2, 3, 2, 5, 1) = the first 5 terms of A014963.
%Y A140254 Cf. A014963, A008683, A140255, A140256.
%Y A140254 Sequence in context: A067458 A088330 A128263 this_sequence A095202 A093443 A099092
%Y A140254 Adjacent sequences: A140251 A140252 A140253 this_sequence A140255 A140256 A140257
%K A140254 nonn
%O A140254 1,3
%A A140254 Gary W. Adamson and Mats Granvik (qntmpkt(AT)yahoo.com), May 16 2008, Jun 29 2008
%E A140254 More terms from Mats Granvik (mgranvik(AT)abo.fi), Jun 29 2008
%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(kRows n=0..50 of triangle, flattened
%H A004174 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, December 1972 [alternative scanned copy].
%H A004174 Eric Weisstein's World of Mathematics, MathWorld: Euler Polynomial
%Y A004174 Cf. A099932
%Y A004174 Sequence in context: A081236 A103328 A114122 this_sequence A049797 A116578 A078050
%Y A004174 Adjacent sequences: A004171 A004172 A004173 this_sequence A004175 A004176 A004177
%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 Sequence in context: A103328 A114122 A004174 this_sequence A116578 A078050 A134271
%Y A049797 Adjacent sequences: A049794 A049795 A049796 this_sequence A049798 A049799 A049800
%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 Sequence in context: A114122 A004174 A049797 this_sequence A078050 A134271 A094403
%Y A116578 Adjacent sequences: A116575 A116576 A116577 this_sequence A116579 A116580 A116581
%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 Sequence in context: A004174 A049797 A116578 this_sequence A134271 A094403 A129760
%Y A078050 Adjacent sequences: A078047 A078048 A078049 this_sequence A078051 A078052 A078053
%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 Sequence in context: A049797 A116578 A078050 this_sequence A094403 A129760 A057377
%Y A134271 Adjacent sequences: A134268 A134269 A134270 this_sequence A134272 A134273 A134274
%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 Sequence in context: A116578 A078050 A134271 this_sequence A129760 A057377 A131772
%Y A094403 Adjacent sequences: A094400 A094401 A094402 this_sequence A094404 A094405 A094406
%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.
%C A129760 Also the number of Ducci sequences with period n.
%D A129760 R. Brown and J. L. Merzel, The number of Ducci sequences with a given period, Fib. Quart., 45 (2007), 115-121.
%F A129760 Equals n - A006519(n). - njas, May 26 2008
%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, A104594, A059991, A006519.
%Y A129760 Sequence in context: A078050 A134271 A094403 this_sequence A057377 A131772 A021493
%Y A129760 Adjacent sequences: A129757 A129758 A129759 this_sequence A129761 A129762 A129763
%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, Table of n, a(n) for n = 0..71 (from link below)
%H A057377 I. Jensen, More terms
%Y A057377 Cf. A057374-A057405.
%Y A057377 Sequence in context: A134271 A094403 A129760 this_sequence A131772 A021493 A084247
%Y A057377 Adjacent sequences: A057374 A057375 A057376 this_sequence A057378 A057379 A057380
%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 Sequence in context: A094403 A129760 A057377 this_sequence A021493 A084247 A070692
%Y A131772 Adjacent sequences: A131769 A131770 A131771 this_sequence A131773 A131774 A131775
%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 Sequence in context: A129760 A057377 A131772 this_sequence A084247 A070692 A091684
%Y A021493 Adjacent sequences: A021490 A021491 A021492 this_sequence A021494 A021495 A021496
%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 Second differences of Jacobsthal numbers, A001045. - Paul Curtz (bpcrtz(AT)free.fr), Jun 30 2008
%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 Sequence in context: A057377 A131772 A021493 this_sequence A070692 A091684 A100050
%Y A084247 Adjacent sequences: A084244 A084245 A084246 this_sequence A084248 A084249 A084250
%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 Sequence in context: A131772 A021493 A084247 this_sequence A091684 A100050 A004482
%Y A070692 Adjacent sequences: A070689 A070690 A070691 this_sequence A070693 A070694 A070695
%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 Sequence in context: A021493 A084247 A070692 this_sequence A100050 A004482 A111677
%Y A091684 Adjacent sequences: A091681 A091682 A091683 this_sequence A091685 A091686 A091687
%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.
%H A100050 Zerinvary Lajos, Sage Notebooks
%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)}.
%o A100050 sage: [lucas_number1(n,2,1)*lucas_number1(n,1,1) for n in xrange(0,88)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 06 2008
%Y A100050 Cf. A099837, A099443, A011655, A100047, A100048, A100051, A091684.
%Y A100050 Sequence in context: A084247 A070692 A091684 this_sequence A004482 A111677 A049271
%Y A100050 Adjacent sequences: A100047 A100048 A100049 this_sequence A100051 A100052 A100053
%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 Sequence in context: A070692 A091684 A100050 this_sequence A111677 A049271 A004178
%Y A004482 Adjacent sequences: A004479 A004480 A004481 this_sequence A004483 A004484 A004485
%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 Sequence in context: A091684 A100050 A004482 this_sequence A049271 A004178 A068333
%Y A111677 Adjacent sequences: A111674 A111675 A111676 this_sequence A111678 A111679 A111680
%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 Sequence in context: A100050 A004482 A111677 this_sequence A004178 A068333 A121451
%Y A049271 Adjacent sequences: A049268 A049269 A049270 this_sequence A049272 A049273 A049274
%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 Sequence in context: A004482 A111677 A049271 this_sequence A068333 A121451 A096984
%Y A004178 Adjacent sequences: A004175 A004176 A004177 this_sequence A004179 A004180 A004181
%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 Sequence in context: A111677 A049271 A004178 this_sequence A121451 A096984 A104601
%Y A068333 Adjacent sequences: A068330 A068331 A068332 this_sequence A068334 A068335 A068336
%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 Sequence in context: A049271 A004178 A068333 this_sequence A096984 A104601 A133144
%Y A121451 Adjacent sequences: A121448 A121449 A121450 this_sequence A121452 A121453 A121454
%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 Sequence in context: A004178 A068333 A121451 this_sequence A104601 A133144 A098123
%Y A096984 Adjacent sequences: A096981 A096982 A096983 this_sequence A096985 A096986 A096987
%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, On the arithmetic product of combinatorial species
%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 Sequence in context: A068333 A121451 A096984 this_sequence A133144 A098123 A066659
%Y A104601 Adjacent sequences: A104598 A104599 A104600 this_sequence A104602 A104603 A104604
%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 Sequence in context: A121451 A096984 A104601 this_sequence A098123 A066659 A085623
%Y A133144 Adjacent sequences: A133141 A133142 A133143 this_sequence A133145 A133146 A133147
%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 Sequence in context: A096984 A104601 A133144 this_sequence A066659 A085623 A002885
%Y A098123 Adjacent sequences: A098120 A098121 A098122 this_sequence A098124 A098125 A098126
%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 Sequence in context: A104601 A133144 A098123 this_sequence A085623 A002885 A011121
%Y A066659 Adjacent sequences: A066656 A066657 A066658 this_sequence A066660 A066661 A066662
%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 Sequence in context: A133144 A098123 A066659 this_sequence A002885 A011121 A117902
%Y A085623 Adjacent sequences: A085620 A085621 A085622 this_sequence A085624 A085625 A085626
%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 Index entries for sequences related to Steiner systems
%Y A002885 Sequence in context: A098123 A066659 A085623 this_sequence A011121 A117902 A021087
%Y A002885 Adjacent sequences: A002882 A002883 A002884 this_sequence A002886 A002887 A002888
%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 Sequence in context: A066659 A085623 A002885 this_sequence A117902 A021087 A120558
%Y A011121 Adjacent sequences: A011118 A011119 A011120 this_sequence A011122 A011123 A011124
%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 Sequence in context: A085623 A002885 A011121 this_sequence A021087 A120558 A120554
%Y A117902 Adjacent sequences: A117899 A117900 A117901 this_sequence A117903 A117904 A117905
%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 Sequence in context: A002885 A011121 A117902 this_sequence A120558 A120554 A120710
%Y A021087 Adjacent sequences: A021084 A021085 A021086 this_sequence A021088 A021089 A021090
%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, Table of n, a(n) for n = 0..238 [from link below]
%H A120558 I. Jensen, More terms
%Y A120558 Sequence in context: A011121 A117902 A021087 this_sequence A120554 A120710 A115780
%Y A120558 Adjacent sequences: A120555 A120556 A120557 this_sequence A120559 A120560 A120561
%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, Table of n, a(n) for n = 0..254 [from link below]
%H A120554 I. Jensen, More terms
%Y A120554 Sequence in context: A117902 A021087 A120558 this_sequence A120710 A115780 A101189
%Y A120554 Adjacent sequences: A120551 A120552 A120553 this_sequence A120555 A120556 A120557
%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 Sequence in context: A021087 A120558 A120554 this_sequence A115780 A101189 A070015
%Y A120710 Adjacent sequences: A120707 A120708 A120709 this_sequence A120711 A120712 A120713
%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 Sequence in context: A120558 A120554 A120710 this_sequence A101189 A070015 A021492
%Y A115780 Adjacent sequences: A115777 A115778 A115779 this_sequence A115781 A115782 A115783
%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 Sequence in context: A120554 A120710 A115780 this_sequence A070015 A021492 A077119
%Y A101189 Adjacent sequences: A101186 A101187 A101188 this_sequence A101190 A101191 A101192
%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, Table of n, a(n) for n = 1..9884
%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 Sequence in context: A120710 A115780 A101189 this_sequence A021492 A077119 A002938
%Y A070015 Adjacent sequences: A070012 A070013 A070014 this_sequence A070016 A070017 A070018
%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 Sequence in context: A115780 A101189 A070015 this_sequence A077119 A002938 A111938
%Y A021492 Adjacent sequences: A021489 A021490 A021491 this_sequence A021493 A021494 A021495
%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)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 Sequence in context: A021492 A077119 A002938 this_sequence A055978 A069025 A066442
%Y A111938 Adjacent sequences: A111935 A111936 A111937 this_sequence A111939 A111940 A111941
%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 Sequence in context: A077119 A002938 A111938 this_sequence A069025 A066442 A086134
%Y A055978 Adjacent sequences: A055975 A055976 A055977 this_sequence A055979 A055980 A055981
%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 Sequence in context: A002938 A111938 A055978 this_sequence A066442 A086134 A071090
%Y A069025 Adjacent sequences: A069022 A069023 A069024 this_sequence A069026 A069027 A069028
%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 Sequence in context: A111938 A055978 A069025 this_sequence A086134 A071090 A105221
%Y A066442 Adjacent sequences: A066439 A066440 A066441 this_sequence A066443 A066444 A066445
%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 Sequence in context: A055978 A069025 A066442 this_sequence A071090 A105221 A061376
%Y A086134 Adjacent sequences: A086131 A086132 A086133 this_sequence A086135 A086136 A086137
%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 Sequence in context: A069025 A066442 A086134 this_sequence A105221 A061376 A058974
%Y A071090 Adjacent sequences: A071087 A071088 A071089 this_sequence A071091 A071092 A071093
%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, Table of n, a(n) for n=1..1000
%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 Sequence in context: A066442 A086134 A071090 this_sequence A061376 A058974 A019962
%Y A105221 Adjacent sequences: A105218 A105219 A105220 this_sequence A105222 A105223 A105224
%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 Sequence in context: A086134 A071090 A105221 this_sequence A058974 A019962 A086131
%Y A061376 Adjacent sequences: A061373 A061374 A061375 this_sequence A061377 A061378 A061379
%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 Sequence in context: A071090 A105221 A061376 this_sequence A019962 A086131 A104755
%Y A058974 Adjacent sequences: A058971 A058972 A058973 this_sequence A058975 A058976 A058977
%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 Sequence in context: A105221 A061376 A058974 this_sequence A086131 A104755 A054013
%Y A019962 Adjacent sequences: A019959 A019960 A019961 this_sequence A019963 A019964 A019965
%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 Sequence in context: A061376 A058974 A019962 this_sequence A104755 A054013 A048050
%Y A086131 Adjacent sequences: A086128 A086129 A086130 this_sequence A086132 A086133 A086134
%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 Sequence in context: A058974 A019962 A086131 this_sequence A054013 A048050 A078153
%Y A104755 Adjacent sequences: A104752 A104753 A104754 this_sequence A104756 A104757 A104758
%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 Sequence in context: A019962 A086131 A104755 this_sequence A048050 A078153 A104035
%Y A054013 Adjacent sequences: A054010 A054011 A054012 this_sequence A054014 A054015 A054016
%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, Table of n, a(n) for n=1..10000
%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 Sequence in context: A086131 A104755 A054013 this_sequence A078153 A104035 A115333
%Y A048050 Adjacent sequences: A048047 A048048 A048049 this_sequence A048051 A048052 A048053
%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 Sequence in context: A104755 A054013 A048050 this_sequence A104035 A115333 A105523
%Y A078153 Adjacent sequences: A078150 A078151 A078152 this_sequence A078154 A078155 A078156
%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 Sequence in context: A054013 A048050 A078153 this_sequence A115333 A105523 A126120
%Y A104035 Adjacent sequences: A104032 A104033 A104034 this_sequence A104036 A104037 A104038
%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= 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
%C A126120 Essentially the same as A097331. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 15 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, Hyperelliptic curves, L-polynomials, and random matrices.
%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 Sequence in context: A104035 A115333 A105523 this_sequence A090192 A097331 A094032
%Y A126120 Adjacent sequences: A126117 A126118 A126119 this_sequence A126121 A126122 A126123
%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 Sequence in context: A115333 A105523 A126120 this_sequence A097331 A094032 A117780
%Y A090192 Adjacent sequences: A090189 A090190 A090191 this_sequence A090193 A090194 A090195
%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.
%C A097331 Essentially the same as A126120. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 15 2008
%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 Sequence in context: A105523 A126120 A090192 this_sequence A094032 A117780 A082974
%Y A097331 Adjacent sequences: A097328 A097329 A097330 this_sequence A097332 A097333 A097334
%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, Minimum braids: a complete invariant of knots and links.
%Y A094032 Cf. A094029, A094030, A094031.
%Y A094032 Sequence in context: A126120 A090192 A097331 this_sequence A117780 A082974 A066283
%Y A094032 Adjacent sequences: A094029 A094030 A094031 this_sequence A094033 A094034 A094035
%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: Palindromic Prime.
%Y A117780 Cf. A029973, A117700.
%Y A117780 Sequence in context: A090192 A097331 A094032 this_sequence A082974 A066283 A014842
%Y A117780 Adjacent sequences: A117777 A117778 A117779 this_sequence A117781 A117782 A117783
%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 Sequence in context: A097331 A094032 A117780 this_sequence A066283 A014842 A132816
%Y A082974 Adjacent sequences: A082971 A082972 A082973 this_sequence A082975 A082976 A082977
%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 Sequence in context: A094032 A117780 A082974 this_sequence A014842 A132816 A077453
%Y A066283 Adjacent sequences: A066280 A066281 A066282 this_sequence A066284 A066285 A066286
%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 Sequence in context: A117780 A082974 A066283 this_sequence A132816 A077453 A021491
%Y A014842 Adjacent sequences: A014839 A014840 A014841 this_sequence A014843 A014844 A014845
%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 Sequence in context: A082974 A066283 A014842 this_sequence A077453 A021491 A121705
%Y A132816 Adjacent sequences: A132813 A132814 A132815 this_sequence A132817 A132818 A132819
%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 transfer matrix approach to the enumeration of plane meanders, J. Phys. A 33 (2000), no. 34, 5953-5963.
%e A077453 2.05319873277447561012612863...
%Y A077453 Cf. A077452.
%Y A077453 Sequence in context: A066283 A014842 A132816 this_sequence A021491 A121705 A071782
%Y A077453 Adjacent sequences: A077450 A077451 A077452 this_sequence A077454 A077455 A077456
%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 Sequence in context: A014842 A132816 A077453 this_sequence A121705 A071782 A107363
%Y A021491 Adjacent sequences: A021488 A021489 A021490 this_sequence A021492 A021493 A021494
%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=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 Sequence in context: A077453 A021491 A121705 this_sequence A107363 A095245 A086280
%Y A071782 Adjacent sequences: A071779 A071780 A071781 this_sequence A071783 A071784 A071785
%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 Sequence in context: A021491 A121705 A071782 this_sequence A095245 A086280 A083714
%Y A107363 Adjacent sequences: A107360 A107361 A107362 this_sequence A107364 A107365 A107366
%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 Sequence in context: A121705 A071782 A107363 this_sequence A086280 A083714 A137421
%Y A095245 Adjacent sequences: A095242 A09524