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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 September 23 16:41 EDT 2020. Contains 337315 sequences. (Running on oeis4.)