A176971 Expansion of (1+x)/(1+x-x^3) in powers of x.

1, 0, 0, 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

Offset: 0

Author

Roger L. Bagula, Apr 29 2010

Keywords

• sign
• easy

Comments

Except for signs the sequence is the essentially same as A078013, A050935 and A104769.

Padovan sequence extended to negative indices. - Hugo Pfoertner, Jul 16 2017

Examples

			G.f. = 1 + x^3 - x^4 + x^5 - x^7 + 2*x^8 - 2*x^9 + x^10 + x^11 - 3*x^12 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Taras Goy and Mark Shattuck, Toeplitz-Hessenberg determinant formulas for the sequence F_n-1, Online J. Anal. Comb. (2025) Vol. 19, Paper 1, 1-26. See p. 12.
• Yüksel Soykan, Summing Formulas For Generalized Tribonacci Numbers, arXiv:1910.03490 [math.GM], 2019.
• Wikipedia, Padovan sequence.
• Index entries for linear recurrences with constant coefficients, signature (-1,0,1). [From _R. J. Mathar_, Apr 30 2010]

Crossrefs

Cf. A000931, A005251, A050935, A078013, A104769, A182097.

Programs

Magma

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x)/(1+x-x^3))); // G. C. Greubel, Sep 25 2018

Mathematica

a[0] := 1; a[1] = 0; a[2] = 0;
a[n_] := a[n] = a[n - 2] + a[n - 3];
b = Table[a[n], {n, 0, 50}];
Table[b[[n]]^2 - b[[n - 1]]*b[[n + 1]], {n, 1, Length[b] - 1}]
a[ n_] := If[ n >= 0, SeriesCoefficient[ (1 + x) / (1 + x - x^3), {x, 0, n}], SeriesCoefficient[ 1 / (1 - x^2 - x^3), {x, 0, Abs@n}]]; (* Michael Somos, Dec 13 2013 *)

PARI

{a(n) = if( n>=0, polcoeff( (1 + x) / (1 + x - x^3) + x * O(x^n), n), polcoeff( 1 / (1 - x^2 - x^3) + x * O(x^-n), -n))}; /* Michael Somos, Dec 13 2013 */

Formula

a(n) = A000931(n)^2 -A000931(n-1)*A000931(n+1).

a(n) = -a(n-1) +a(n-3). - R. J. Mathar, Apr 30 2010

a(n) = -A104769(n) - A104769(n+1). - Ralf Stephan, Aug 18 2013

G.f.: 1 / (1 - x^3 / (1 + x)). - Michael Somos, Dec 13 2013

a(n) = A182097(-n) for all n in Z. - Michael Somos, Dec 13 2013

A000931(n) = a(n)^2 - a(n-1) * a(n+1). - Michael Somos, Dec 13 2013

Binomial transform is A005251(n+1). - Michael Somos, Dec 13 2013

Extensions

Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021