A247917 Expansion of 1 / (1 + x - x^3) in powers of x.

1, -1, 1, 0, -1, 2, -2, 1, 1, -3, 4, -3, 0, 4, -7, 7, -3, -4, 11, -14, 10, 1, -15, 25, -24, 9, 16, -40, 49, -33, -7, 56, -89, 82, -26, -63, 145, -171, 108, 37, -208, 316, -279, 71, 245, -524, 595, -350, -174, 769, -1119, 945, -176, -943, 1888, -2064, 1121

Offset: 0

Author

Michael Somos, Sep 26 2014

Keywords

• sign
• easy

Examples

			G.f. = 1 - x + x^2 - x^4 + 2*x^5 - 2*x^6 + x^7 + x^8 - 3*x^9 + 4*x^10 + ...
		

Links

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

Crossrefs

Cf. A000931, A050935, A078013, A104769, A176971.

Programs

Magma

m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1 + x - x^3)));  // G. C. Greubel, Aug 04 2018

Mathematica

CoefficientList[Series[1/(1 + x - x^3), {x, 0, 100}], x] (* Vincenzo Librandi, Sep 27 2014 *)
LinearRecurrence[{-1,0,1},{1,-1,1},60] (* Harvey P. Dale, Apr 10 2025 *)

PARI

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

Formula

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

0 = a(n) - a(n+2) - a(n+3) for all n in Z.

a(-n) = A000931(n) for all n in Z.

a(n) = A176971(n+3) for all n in Z.

-a(n) = A104769(n+1) for all n in Z.

(-1)^n * a(n) = A050935(n+3) for all n in Z.

-(-1)^n * a(n) = A078013(n+3) for all n in Z.