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A111927
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Expansion of x^3 / ((x-1)*(2*x-1)*(x^2-x+1)).
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8
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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
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OFFSET
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0,5
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COMMENTS
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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
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LINKS
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Antoine-Augustin Cournot, Solution d'un problème d'analyse combinatoire, Bulletin des Sciences Mathématiques, Physiques et Chimiques, item 34, volume 11, 1829, pages 93-97. Also at Google Books. Page 97 case p=3 formula y^(0) = a(n). (But misprint "- (2/3)*cos" should be "+ (2/3)*cos".)
Christian Ramus, Solution générale d'un problème d'analyse combinatoire, Journal für die Reine und Angewandte Mathematik (Crelle's journal), volume 11, 1834, pages 353-355. Page 353 case p=3 formula y^(0) = a(n). (But misprint "+ (1/3)*cos" should be "+ (2/3)*cos", per the general case equation A page 354.)
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FORMULA
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a(n+2) - a(n+1) + a(n) = A000225(n).
a(n) = 2^n/3 + 2*cos((Pi*n)/3)/3 - 1. [Cournot]
a(n) = 4*a(n-1) - 6*a(n-2) + 5*a(n-3) - 2*a(n-4) for n > 3.
(End)
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MAPLE
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seq(sum(binomial(n, k*3), k=1..n), n=0..33); # Zerinvary Lajos), Oct 23 2007
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MATHEMATICA
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LinearRecurrence[{4, -6, 5, -2}, {0, 0, 0, 1}, 40] (* Harvey P. Dale, Jul 04 2017 *)
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PROG
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(PARI) concat(vector(3), Vec(x^3/((x-1)*(2*x-1)*(x^2-x+1)) + O(x^40))) \\ Colin Barker, Feb 10 2017
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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