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%I #23 Oct 25 2020 12:36:00
%S 1,0,0,2,-1,0,4,-4,1,8,-12,6,15,-32,24,24,-79,80,24,-182,239,-32,-388,
%T 660,-303,-744,1708,-1266,-1185,4160,-4240,-1104,9505,-12640,2032,
%U 20114,-34785,16704,38196,-89684,68193,59688,-217564,226070,51183,-494816,669704,-123704,-1040815,1834224
%N Expansion of (1-x)^(-1)/(1+x+x^2-x^3).
%H Alois P. Heinz, <a href="/A077908/b077908.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,2,-1).
%F a(n) = -1 * term (4,1) in the 4x4 matrix [0,0,1,0; 1,0,-1,0; 0,1,-1,0; 0,0,-1,1]^(n+3) - _Alois P. Heinz_, Jul 24 2008
%F a(n) = -A008937(-n-3). - _Alois P. Heinz_, Jul 24 2008
%F G.f.: 1 / (1 - 2*x^3 + x^4). - _Michael Somos_, Aug 19 2014
%F a(n) = -a(n-1) - a(n-2) + a(n-3) + 1 = 2*a(n-3) - a(n-4) for all n in Z. - _Michael Somos_, Aug 19 2014
%F a(n) - a(n-1) = A057597(n+2). (first differences). - _R. J. Mathar_, Oct 16 2017
%e G.f. = 1 + 2*x^3 - x^4 + 4*x^6 - 4*x^7 + x^8 + 8*x^9 - 12*x^10 + 6*x^11 + ...
%p a:= n-> -(<<0|0|1|0>, <1|0|-1|0>, <0|1|-1|0>, <0|0|-1|1>>^(n+3))[4, 1]:
%p seq(a(n), n=0..50); # _Alois P. Heinz_, Jul 24 2008
%t a[n_] := -MatrixPower[{{0, 0, 1, 0}, {1, 0, -1, 0}, {0, 1, -1, 0}, {0, 0, -1, 1}}, n+3][[4, 1]]; Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Feb 14 2016, after _Alois P. Heinz_ *)
%t LinearRecurrence[{0,0,2,-1},{1,0,0,2},50] (* _Harvey P. Dale_, Oct 25 2020 *)
%o (PARI) Vec((1-x)^(-1)/(1+x+x^2-x^3)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 26 2012
%o (PARI) {a(n) = if( n<0, polcoeff( - x^4 / (1 - 2*x + x^4) + x * O(x^-n), -n), polcoeff( 1 / (1 - 2*x^3 + x^4) + x * O(x^n), n))}; /* _Michael Somos_, Aug 19 2014 */
%Y Cf. A008937.
%K sign,easy
%O 0,4
%A _N. J. A. Sloane_, Nov 17 2002
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