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A077961 Expansion of 1 / (1 + x^2 - x^3) in powers of x.

%I A077961 #50 Apr 13 2025 07:33:47
%S A077961 1,0,-1,1,1,-2,0,3,-2,-3,5,1,-8,4,9,-12,-5,21,-7,-26,28,19,-54,9,73,
%T A077961 -63,-64,136,1,-200,135,201,-335,-66,536,-269,-602,805,333,-1407,472,
%U A077961 1740,-1879,-1268,3619,-611,-4887,4230,4276,-9117,-46,13393,-9071,-13439,22464,4368,-35903,18096,40271,-53999
%N A077961 Expansion of 1 / (1 + x^2 - x^3) in powers of x.
%H A077961 Alois P. Heinz, <a href="/A077961/b077961.txt">Table of n, a(n) for n = 0..1000</a>
%H A077961 N. Gogin and A. Mylläri, <a href="http://math.unm.edu/~aca/ACA/2013/Nonstandard/Gogin.pdf">Padovan-like sequences and Bell polynomials</a>, Proceedings of Applications of Computer Algebra ACA, 2013.
%H A077961 Michael D. Hirschhorn, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/43-4/paper43-4-5.pdf">Non-trivial intertwined second-order recurrence relations</a>, Fibonacci Quart. 43 (2005), no. 4, 316-325. See L_n.
%H A077961 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,-1,1).
%F A077961 a(n) = Sum_{k=0..floor(n/2)} (-1)^(n-k)*binomial(k, n-2*k). - _Paul Barry_, Jun 24 2005
%F A077961 From _Alois P. Heinz_, Jun 20 2008: (Start)
%F A077961 a(n) = term (1,1) in matrix [0,1,0; -1,0,1; 1,0,0]^n.
%F A077961 a(n) = A000930 (-3-n). (End)
%F A077961 a(-n) = A078012(n). - _Michael Somos_, May 03 2011
%F A077961 From _Michael Somos_, Jan 08 2014: (Start)
%F A077961 a(-n) = A135851(n+2).
%F A077961 a(n)^2 - a(n-1)*a(n+1) = A135851(n+5). (End)
%F A077961 G.f.: Q(0)/2 , where Q(k) = 1 + 1/(1 - x^2*(4*k+1 - x )/( x^2*(4*k+3 - x ) - 1/Q(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Sep 09 2013
%F A077961 a(n)^2 + a(n-1)^2 + a(n-6)^2 = a(n-2)^2 + 3*a(n-3)^2 + a(n-4)^2 + a(n-5)^2. - _Greg Dresden_, Apr 12 2025
%e A077961 G.f. = 1 - x^2 + x^3 + x^4 - 2*x^5 + 3*x^7 - 2*x^8 - 3*x^9 + 5*x^10 + x^11 + ...
%p A077961 a:= n-> (<<0|1|0>, <-1|0|1>, <1|0|0>>^n)[1, 1]:
%p A077961 seq(a(n), n=0..50);  # _Alois P. Heinz_, Jun 20 2008
%t A077961 a[ n_] := If[ n >= 0, SeriesCoefficient[ 1 / (1 + x^2 - x^3), {x, 0, n}], SeriesCoefficient[ x^3 / (1 - x - x^3), {x, 0, -n}]] (* _Michael Somos_, Jan 08 2014 *)
%o A077961 (PARI) {a(n) = if( n<0, polcoeff( x^3 / (1 - x - x^3) + x * O(x^-n), -n), polcoeff( 1 / (1 + x^2 - x^3) + x * O(x^n), n))} /* _Michael Somos_, Jan 08 2014 */
%o A077961 (Magma) I:=[1,0,-1]; [n le 3 select I[n] else -Self(n-2) +Self(n-3): n in [1..70]]; // _G. C. Greubel_, Mar 28 2024
%o A077961 (SageMath) [sum((-1)^(n+k)*binomial(k,n-2*k) for k in range(1+n//2)) for n in range(71)] # _G. C. Greubel_, Mar 28 2024
%Y A077961 Cf. A000930, A078012, A135851.
%K A077961 sign,easy
%O A077961 0,6
%A A077961 _N. J. A. Sloane_, Nov 17 2002