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 A111927 Expansion of x^3 / ((x-1)*(2*x-1)*(x^2-x+1)). 5
 0, 0, 0, 1, 4, 10, 21, 42, 84, 169, 340, 682, 1365, 2730, 5460, 10921, 21844, 43690, 87381, 174762, 349524, 699049, 1398100, 2796202, 5592405, 11184810, 22369620, 44739241, 89478484, 178956970, 357913941, 715827882, 1431655764, 2863311529, 5726623060 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Binomial transform of sequence (0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0). Note: the binomial transform of the sequence (0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0) is A111926; the binomial transform of the sequence (0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0) is A024495 (disregarding first two terms, which are both zero). The sequence relates the calculation of the logarithm of the Twin Prime Constants of order 3 to the sequence of prime zeta functions, see definition 7 in arXiv:0903.2514. - R. J. Mathar, Mar 28 2009 LINKS Colin Barker, Table of n, a(n) for n = 0..1000 R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arXiv:0903.2514 [math.NT], 2009-2011. Index entries for linear recurrences with constant coefficients, signature (4,-6,5,-2). FORMULA a(n+2) - a(n+1) + a(n) = A000225(n). a(n) - a(n-1)= A024495(n-1). From Colin Barker, Feb 10 2017: (Start) a(n) = 2^n/3 + 2*cos((Pi*n)/3)/3 - 1. a(n) = 4*a(n-1) - 6*a(n-2) + 5*a(n-3) - 2*a(n-4) for n > 3. (End) a(n) = (2^n+A087204(n))/3 - 1. - R. J. Mathar, Aug 07 2017 MAPLE seq(sum(binomial(n, k*3), k=1..n), n=0..33); # Zerinvary Lajos), Oct 23 2007 MATHEMATICA LinearRecurrence[{4, -6, 5, -2}, {0, 0, 0, 1}, 40] (* Harvey P. Dale, Jul 04 2017 *) PROG (PARI) concat(vector(3), Vec(x^3/((x-1)*(2*x-1)*(x^2-x+1)) + O(x^40))) \\ Colin Barker, Feb 10 2017 CROSSREFS Cf. A000295, A111926, A024495. Sequence in context: A132925 A264079 A053643 * A329361 A290998 A227803 Adjacent sequences:  A111924 A111925 A111926 * A111928 A111929 A111930 KEYWORD easy,nonn AUTHOR Creighton Dement, Aug 21 2005 STATUS approved

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Last modified January 27 04:57 EST 2020. Contains 331291 sequences. (Running on oeis4.)