OFFSET
0,9
COMMENTS
Padovan sequence extended to negative indices. - Hugo Pfoertner, Jul 16 2017
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..5000
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]
FORMULA
a(n) = -a(n-1) +a(n-3). - R. J. Mathar, Apr 30 2010
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
EXAMPLE
G.f. = 1 + x^3 - x^4 + x^5 - x^7 + 2*x^8 - 2*x^9 + x^10 + x^11 - 3*x^12 + ...
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 *)
PROG
(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 */
(Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x)/(1+x-x^3))); // G. C. Greubel, Sep 25 2018
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Roger L. Bagula, Apr 29 2010
EXTENSIONS
Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021
STATUS
approved