Search: id:A026741 Results 1-1 of 1 results found. %I A026741 %S A026741 0,1,1,3,2,5,3,7,4,9,5,11,6,13,7,15,8,17,9,19,10,21,11,23,12,25,13,27, 14, %T A026741 29,15,31,16,33,17,35,18,37,19,39,20,41,21,43,22,45,23,47,24,49,25,51, 26, %U A026741 53,27,55,28,57,29,59,30,61,31,63,32,65,33,67,34,69,35,71,36,73,37,75, 38 %N A026741 a(n) = n if n odd, n/2 if n even. %C A026741 a(n) is the size of largest conjugacy class in D_2n, the dihedral group with 2n elements. - Sharon Sela (sharonsela(AT)hotmail.com), May 14 2002 %C A026741 a(n+1) is the composition length of the n-th symmetric power of the natural representation of a finite subgroup of SL(2,C) of type D_4 (quaternion group). - Paul Boddington (psb(AT)maths.warwick.ac.uk), Oct 23 2003 %C A026741 For n>1 a(n) = greatest common divisor of all permutations of {0,1,..., n} treated as base n+1 integers. - David J. Scambler (dscambler(AT)bmm.com), Nov 08 2006 %C A026741 a(n) is the second principal diagonal of array with rows 1) A005563, 2) (first bisection of A061037)=A142705, 3) (first trisection of A061039)=A144454, 4) first quadrisection of A061041, 5) first quintisection of A061043, 6) first hexasection A061045, 7)first heptasection of A061047, 8) first octosection of A061049, 9) , 10) , 11) , . From Rydberg-Ritz denominators of spectra of hydrogen atom: 1) 0, 3, 8, 15, 24, 35, 48, 63, 80, 99, 120, 143, 2) 0, 3, 2, 15, 6, 35, 12, 63, 20, 99, 39, 143, 3) 0, 1, 8, 5, 8, 35, 16, 7, 80, 11, 40, 143, 4) 0, 3, 1, 15, 3, 35, 3, 63, 5, 99, 15, 143, 5) 0, 3, 8, 3, 24, 7, 48, 63, 16, 99, 6) 0, 1, 2, 5, 2, 35, 4, 7, 20, 7) 0, 3, 8, 15, 24, 5, 48, 9, 8) 0, 3, 1, 15, 3, 35, 3, 63, 5, 99, 15, 143, 9) 0, 1, 8, 5, 8, 35, 16, 7, 80, 11, 40, 143, 10) 0, 3, 2, 3, 6, 7, 12, 63, 4, 99, 6, 11) 0, 3, 8, 15, 24, 35, 48, 63, 80, 9, 120, 13, Thanks to Richard Mathar, April 29, for last four rows. Note first upper principal diagonal 3, 2, 5, 3, 7, 4, 9, =A026741(n+3). [From Paul Curtzz (bpcrtz(AT)free.fr), Sep 13 2009] %C A026741 a(n) = A167192(n+2,2). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 30 2009] %C A026741 Contribution from Paul Curtz (bpcrtz(AT)free.fr), Nov 19 2009: (Start) The array of a(n) and its higher order differences shows essentially 0,0 followed by -3*A001792(.) on the main diagonal: %C A026741 0, 1, 1, 3, 2, 5, 3, 7, 4, 9, 5, 11, 6, 13, 7, 15, 8, 17, 9, 19, 10, 21,... %C A026741 1, 0, 2, -1, 3, -2, 4, -3, 5, -4, 6, -5, 7, -6, 8, -7, 9, -8, 10, -9, 11,... %C A026741 -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, ... %C A026741 3, -5, 7, -9, 11, -13, 15, -17, 19, -21, 23, -25, 27, -29, 31, -33, 35, ... %C A026741 -8, 12, -16, 20, -24, 28, -32, 36, -40, 44, -48, 52, -56, 60, -64, 68, ... %C A026741 20, -28, 36, -44, 52, -60, 68, -76, 84, -92, 100, -108, 116, -124, 132, ... (END) %C A026741 Starting with 1 = triangle A115359 * [1, 2, 3,...]. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 27 2009] %D A026741 Y. Ito, I. Nakamura, Hilbert schemes and simple singularities, New trends in algebraic geometry (Warwick, 1996), 151-233, Cambridge University Press, 1999. %H A026741 L. Euler, De mirabilibus proprietatibus numerorum pentagonalium, par. 2 %H A026741 L. Euler, On the remarkable properties of the pentagonal numbers %H A026741 M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9. %H A026741 Eric Weisstein's World of Mathematics, Simplex Simplex Picking %H A026741 Index entries for sequences related to linear recurrences with constant coefficients %F A026741 G.f.: (x^3+x^2+x)/(1-x^2)^2 - Len Smiley (smiley(AT)math.uaa.alaska.edu), Apr 30 2001 %F A026741 a(n) = n * 2^((n mod 2) - 1) - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 16 2001 %F A026741 a(n) = 2*n/(3+(-1)^n) - Benoit Cloitre (benoit7848c(AT)orange.fr), Mar 24 2002 %F A026741 Multiplicative with a(2^e) = 2^(e-1) and a(p^e) = p^e, p>2. - Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 05 2002 %F A026741 a(n) = n / gcd(n, 2). a(n)/A04589(n) = n/((n+1)(n+2)). %F A026741 For n>1, a(n) = denominator of sum{2/(i*(i+1))|1<=i<=n-1}, numerator=A026741. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jul 25 2002 %F A026741 For n > 1, a(n) = GCD of the n-th and (n-1)th triangular numbers (A000217). - Ross La Haye (rlahaye(AT)new.rr.com), Sep 13 2003 %F A026741 Euler transform of finite sequence [1, 2, -1]. - Michael Somos Jun 15 2005 %F A026741 G.f.: x(1-x^3)/((1-x)(1-x^2)^2) = Sum_{k>0} k(x^k-x^(2k)). - Michael Somos Jun 15 2005 %F A026741 a(n)a(n+3) = - 1 + a(n+1)a(n+2). a(-n)=-a(n). %F A026741 a(n) = Abs[ Numerator[ Det[ DiagonalMatrix[ Table[ 1/i^2 -1, {i, 1, n-1} ] ] + 1 ] ] for n>1. - Alexander Adamchuk (alex(AT)kolmogorov.com), Jun 02 2006 %F A026741 For n > 1, a(n) is the numerator of the average of 1,2,...,n-1; i.e., numerator of A000217(n-1)/(n-1), with corresponding denominators [1,2,1,2,...] (A000034). - Rick L. Shepherd (rshepherd2(AT)hotmail.com), Jun 05 2006 %F A026741 Equals A126988 * (1, -1, 0, 0, 0,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 17 2007 %F A026741 For n >= 1, a(n) = GCD(n,A000217(n)). - Rick L. Shepherd (rshepherd2(AT)hotmail.com), Sep 12 2007 %F A026741 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 18 2009: (Start) %F A026741 a(n) = numer(n/(2*n-2)) for n =>2; A022998(n-1) = denom(n/(2*n-2)) for n =>2. %F A026741 (End) %F A026741 a(n+1)-a(n) = (-1)^n*A028242(n) (first differences). a(n+2)-2*a(n+1)+a(n) = (-1)^(n+1)*(n+1). (2nd differences). a(n+3)-3*a(n+2)+3*a(n+1)-a(n) = (-1)^n*A144396(n+1). (3rd differences). [Paul Curtz (bpcrtz(AT)free.fr), Nov 19 2009] %F A026741 a(A131577(n)) = A166444(n). [Paul Curtz (bpcrtz(AT)free.fr), Nov 19 2009] %p A026741 a:=n->add(2+add((-1)^j, j=1..n),j=2..n):seq(a(n)/2,n=1..74);# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 13 2008] %t A026741 Numerator[Abs[Table[ Det[ DiagonalMatrix[ Table[ 1/i^2 -1, {i, 1, n-1} ] ] + 1 ], {n, 1, 20} ]]] - Alexander Adamchuk (alex(AT)kolmogorov.com), Jun 02 2006 %o A026741 (PARI) a(n)=if(n==0, 0, n/gcd(n,2)) /* Michael Somos Jun 15 2005 */ %o A026741 (PARI) a(n) = numerator(n/2) /* Rick Shepherd Sep 12 2007 */ - Rick L. Shepherd (rshepherd2(AT)hotmail.com), Sep 12 2007 %o A026741 (Other) sage: [lcm(n,2)/2for n in xrange(0, 77)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 07 2009] %Y A026741 Signed version is in A030640. Partial sums give A001318. %Y A026741 Cf. this sequence, A051176, A060819, A060791, A060789 for n / gcd(n, k) with k=2..6. %Y A026741 Cf. A045896, A022998, A060762. %Y A026741 Cf. A126988. %Y A026741 Cf. A109007, A130334. %Y A026741 Cf. A109043 [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 13 2008] %Y A026741 Contribution from Dimitrios Choussos (choussos(AT)yahoo.de), May 11 2009: (Start) %Y A026741 Sequence A075888 and the above sequence are fitting together. %Y A026741 First 2 entries of Sequence A026741 have to be taken out. %Y A026741 In some cases two three or more sequenced entries of A026741 have to be added together to get the next entry of A075888. %Y A026741 Example: Sequences begin with 1,3,2,5,3,7,4,9 (4+9 = 13 next entry in A075888. %Y A026741 But it works out well up to prime around 50.000 (havent tested higher ones). %Y A026741 As A075888 gives a very regular graph. There seems to be a regularity in the primes. (End) %Y A026741 Sequence in context: A076605 A030640 A145051 this_sequence A105658 A083242 A111618 %Y A026741 Adjacent sequences: A026738 A026739 A026740 this_sequence A026742 A026743 A026744 %Y A026741 Cf. A115359 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 27 2009] %K A026741 nonn,easy,nice,frac,mult,new %O A026741 0,4 %A A026741 J. Carl Bellinger (carlb(AT)ctron.com) %E A026741 More terms from David W. Wilson (davidwwilson(AT)comcast.net); better description from Jud McCranie (j.mccranie(AT)comcast.net) %E A026741 Edited by Ralf Stephan (ralf(AT)ark.in-berlin.de), Jun 04 2003 Search completed in 0.002 seconds