A247918 Expansion of (1 + x) / ((1 - x^4) * (1 + x^4 - x^5)) in powers of x.

1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 2, 0, -1, 2, -1, 2, 1, -2, 4, -3, 1, 4, -5, 7, -4, -2, 10, -12, 11, -1, -11, 22, -23, 13, 11, -33, 45, -35, 3, 44, -78, 81, -37, -41, 122, -158, 119, 4, -163, 281, -276, 115, 167, -443, 558, -391, -52, 611, -1000, 949

Offset: 0

Author

Michael Somos, Sep 26 2014

Keywords

• sign
• easy

Examples

			G.f. = 1 + x + x^5 + x^6 + x^8 + x^11 + 2*x^13 - x^15 + 2*x^16 - x^17 + ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,-1,1,-1,2,-2,2,-1).

Crossrefs

Cf. A010892, A056594, A077905, A112553, A176971, A247907.

Programs

Magma

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!((1 + x)/((1-x^4)*(1+x^4-x^5)))); // G. C. Greubel, Aug 04 2018

Mathematica

CoefficientList[Series[(1+x)/((1-x^4)(1+x^4-x^5)), {x, 0, 100}], x] (* Vincenzo Librandi, Sep 27 2014 *)

PARI

{a(n) = if( n<0, n=-8-n; polcoeff( -1/((1-x)*(1-x+x^2)*(1+x^2)*(1 - x^2 - x^3)) + x * O(x^n), n), polcoeff( 1/((1-x)*(1-x+x^2)*(1+x^2)*(1+x-x^3)) + x * O(x^n), n))};

SageMath

def A247918_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)/((1-x^4)*(1+x^4-x^5)) ).list()
A247918_list(70) # G. C. Greubel, Aug 08 2022

Formula

G.f.: 1 / ((1 - x) * (1 - x + x^2) * (1 + x^2) * (1 + x - x^3)).

a(n) = a(n+1) + a(n+5) - mod(floor((n-1)/2), 2) for all n in Z.

a(n) = -A247907(-8-n) for all n in Z.

Convolution of A077905 and A112553.

a(n) = (35 + 7*(A056594(n) + 3*A056594(n-1)) + 10*(3*A010892(n) - A010892(n-1)) - 2*(A176971(n) + 4*A172971(n-1) + 12*A176971(n-2)))/70. - G. C. Greubel, Aug 08 2022