A257543 Expansion of 1 / (1 - x^5 - x^8 + x^9) in powers of x.

1, 0, 0, 0, 0, 1, 0, 0, 1, -1, 1, 0, 0, 2, -2, 1, 1, -2, 4, -3, 1, 3, -6, 7, -3, -2, 9, -13, 11, -1, -11, 22, -23, 12, 10, -33, 46, -35, 2, 43, -78, 81, -37, -41, 122, -159, 118, 4, -162, 281, -277, 114, 167, -443, 558, -391, -52, 610, -1001, 949, -338, -662

Offset: 0

Author

Michael Somos, Apr 28 2015

Keywords

• sign
• easy

Examples

			G.f. = 1 + x^5 + x^8 - x^9 + x^10 + 2*x^13 - 2*x^14 + x^15 + x^16 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2500
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1,0,0,1,-1).

Crossrefs

Cf. A104769, A233522, A247917, A247918.

Programs

Magma

m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1 / ((1-x^4)*(1+x^4-x^5)))); // G. C. Greubel, Aug 02 2018

Mathematica

a[ n_] := SeriesCoefficient[ If[ n >= 0, 1 / (1 - x^5 - x^8 + x^9), -x^9 /(1 - x - x^4 + x^9)], {x, 0, Abs@n}];

PARI

{a(n) = if( n>=0, polcoeff( 1 / (1 - x^5 - x^8 + x^9) + x * O(x^n), n), polcoeff( -x^9 / (1 - x - x^4 + x^9) + x * O(x^-n), -n))};

Formula

G.f.: 1 / ((1 - x^4) * (1 + x^4 - x^5)) = (1 + x) / ((1 + x^3) * (1 - x^4) * (1 + x - x^3)).

a(n) = a(n-5) + a(n-8) - a(n-9) for all n in Z.

a(n) - a(n+2) - a(n+3) has period 12.

a(n) - a(n+12) = A104769(n+5) = -A247917(n+4) for all n in Z.

a(n) + a(n+1) = A247918(n) for all n in Z.

a(n) = -A233522(-9 - n) for all n in Z.