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A139714
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a(n) = Sum_{k>=0} binomial(n,5*k+2).
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12
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0, 0, 1, 3, 6, 10, 15, 22, 36, 72, 165, 385, 859, 1807, 3614, 6995, 13380, 25773, 50559, 101118, 204820, 416405, 843756, 1698458, 3396916, 6765175, 13455325, 26789257, 53457121, 106914242, 214146295, 429124630, 859595529, 1720537327, 3441074654
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OFFSET
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0,4
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COMMENTS
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where M = a 5 X 5 matrix [1,1,0,0,0; 0,1,1,0,0; 0,0,1,1,0; 0,0,0,1,1; 1,0,0,0,1].
Sum of terms = 2^n. Example: M^6 = [7, 15, 20, 15, 7], sum = 2^6 = 64. (End)
{A139398, A133476, A139714, A139748, A139761} is the difference analog of the hyperbolic functions of order 5, {h_1(x), h_2(x), h_3(x), h_4(x), h_5 (x)}. For a definition see [Erdelyi] and the Shevelev link. - Vladimir Shevelev, Jun 18 2017
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REFERENCES
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A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, Chapter XVIII.
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LINKS
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FORMULA
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G.f.: -x^2*(x-1)^2/((2*x-1)*(x^4-2*x^3+4*x^2-3*x+1)). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 12 2009
a(n) = round((2/5)*(2^(n-1)+phi^n*cos(Pi*(n-4)/5))), where phi is the golden ratio, round(x) is the integer nearest to x. - Vladimir Shevelev, Jun 18 2017
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MAPLE
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a:= n-> (Matrix(5, (i, j)-> `if`((j-i) mod 5 in [0, 1], 1, 0))^n)[4, 1]:
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MATHEMATICA
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CoefficientList[Series[x^2 (x - 1)^2/((1 - 2 x) (x^4 - 2 x^3 + 4 x^2 - 3 x + 1)), {x, 0, 50}], x] (* Vincenzo Librandi, Dec 21 2015 *)
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PROG
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(PARI) a(n) = sum(k=0, n\5, binomial(n, 5*k+2)); \\ Michel Marcus, Dec 21 2015
(PARI) x='x+O('x^100); concat([0, 0], Vec(-x^2*(x-1)^2/((2*x-1)*(x^4-2*x^3+4*x^2-3*x+1)))) \\ Altug Alkan, Dec 21 2015
(Magma) [n le 5 select (n-2)*(n-1)/2 else 5*Self(n-1)- 10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+2*Self(n-5): n in [1..40]]; // Vincenzo Librandi, Dec 21 2015
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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