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

%I A176971 #54 Jul 09 2025 11:09:37
%S A176971 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,
%T A176971 -24,9,16,-40,49,-33,-7,56,-89,82,-26,-63,145,-171,108,37,-208,316,
%U A176971 -279,71,245,-524,595
%N A176971 Expansion of (1+x)/(1+x-x^3) in powers of x.
%C A176971 Except for signs the sequence is the essentially same as A078013, A050935 and A104769.
%C A176971 Padovan sequence extended to negative indices. - _Hugo Pfoertner_, Jul 16 2017
%H A176971 G. C. Greubel, <a href="/A176971/b176971.txt">Table of n, a(n) for n = 0..5000</a>
%H A176971 Taras Goy and Mark Shattuck, <a href="https://doi.org/10.61091/ojac19-01">Toeplitz-Hessenberg determinant formulas for the sequence F_n-1</a>, Online J. Anal. Comb. (2025) Vol. 19, Paper 1, 1-26. See p. 12.
%H A176971 Yüksel Soykan, <a href="https://arxiv.org/abs/1910.03490">Summing Formulas For Generalized Tribonacci Numbers</a>, arXiv:1910.03490 [math.GM], 2019.
%H A176971 Wikipedia, <a href="https://en.wikipedia.org/wiki/Padovan_sequence">Padovan sequence.</a>
%H A176971 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-1,0,1). [From _R. J. Mathar_, Apr 30 2010]
%F A176971 a(n) = A000931(n)^2 -A000931(n-1)*A000931(n+1).
%F A176971 a(n) = -a(n-1) +a(n-3). - _R. J. Mathar_, Apr 30 2010
%F A176971 a(n) = -A104769(n) - A104769(n+1). - _Ralf Stephan_, Aug 18 2013
%F A176971 G.f.: 1 / (1 - x^3 / (1 + x)). - _Michael Somos_, Dec 13 2013
%F A176971 a(n) = A182097(-n) for all n in Z. - _Michael Somos_, Dec 13 2013
%F A176971 A000931(n) = a(n)^2 - a(n-1) * a(n+1). - _Michael Somos_, Dec 13 2013
%F A176971 Binomial transform is A005251(n+1). - _Michael Somos_, Dec 13 2013
%e A176971 G.f. = 1 + x^3 - x^4 + x^5 - x^7 + 2*x^8 - 2*x^9 + x^10 + x^11 - 3*x^12 + ...
%t A176971 a[0] := 1; a[1] = 0; a[2] = 0;
%t A176971 a[n_] := a[n] = a[n - 2] + a[n - 3];
%t A176971 b = Table[a[n], {n, 0, 50}];
%t A176971 Table[b[[n]]^2 - b[[n - 1]]*b[[n + 1]], {n, 1, Length[b] - 1}]
%t A176971 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 *)
%o A176971 (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 */
%o A176971 (Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x)/(1+x-x^3))); // _G. C. Greubel_, Sep 25 2018
%Y A176971 Cf. A000931, A005251, A050935, A078013, A104769, A182097.
%K A176971 sign,easy
%O A176971 0,9
%A A176971 _Roger L. Bagula_, Apr 29 2010
%E A176971 Deleted certain dangerous or potentially dangerous links. - _N. J. A. Sloane_, Jan 30 2021