A117450 Expansion of (1-x+x^2+x^5)/((1-x)^2*(1-x^5)).

1, 1, 2, 3, 4, 7, 9, 12, 15, 18, 23, 27, 32, 37, 42, 49, 55, 62, 69, 76, 85, 93, 102, 111, 120, 131, 141, 152, 163, 174, 187, 199, 212, 225, 238, 253, 267, 282, 297, 312, 329, 345, 362, 379, 396, 415, 433, 452, 471, 490, 511

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-2,1).

Formula

a(n) = 2*a(n-1) - a(n-2) + a(n-5) - 2*a(n-6) + a(n-7).

a(n) = Sum_{k=0..n} binomial(n-k, L(k/5))}, where L(j/p) is the Legendre symbol of j and p.

a(n) = ((5 - sqrt(5))/50)*cos(2*Pi*(2*n+1)/5) + sqrt(((5 +sqrt(5))/50)*sin(2*Pi*(2*n+1)/5) - ((5 + sqrt(5))/50)*cos(Pi*(2*n+1)/5) + sqrt(((5 -sqrt(5))/50)*sin(Pi*(2*n+1)/5) + (n^2 + n + 3)/5.

Mathematica

CoefficientList[Series[(1-x+x^2+x^5)/((1-x)^2(1-x^5)), {x, 0, 80}], x] (* Harvey P. Dale, Nov 22 2018 *)

Prog

(Magma) R<x>:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (1-x+x^2+x^5)/((1-x)^2*(1-x^5)) )); // G. C. Greubel, Jun 03 2021
(Sage)
def A117450_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( (1-x+x^2+x^5)/((1-x)^2*(1-x^5)) ).list()
A117450_list(60) # G. C. Greubel, Jun 03 2021

Crossrefs

Cf. A117451 (first differences), A117452 (second differences).

Sequence in context: A191989 A084913 A270839 * A132381 A073152 A325092

Adjacent sequences: A117447 A117448 A117449 * A117451 A117452 A117453

Keyword

easy,nonn

Author

Paul Barry, Mar 16 2006

Status

approved