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%I #23 Sep 08 2022 08:45:51
%S 0,0,0,0,0,1,0,1,0,1,1,0,2,-1,3,-1,2,1,-1,5,-5,8,-6,6,0,-4,13,-18,25,
%T -24,21,-7,-10,36,-59,81,-87,78,-41,-17,99,-185,264,-304,288,-188,4,
%U 261,-564,853,-1040,1045,-783,220,634,-1673,2719
%N Expansion of x^5/((1-x)*(1+x-x^5)).
%F a(n) = first component of v(n) = M^n v(0), where v(0)=[0,0,0,0,0,1] and
%F M = {{0, 1, 0, 0, 0, 0},
%F {0, 0, 1, 0, 0, 0},
%F {0, 0, 0, 1, 0, 0},
%F {0, 0, 0, 0, 1, 0},
%F {0, 0, 0, 0, 0, 1},
%F {-1, 1, 0, 0, 1, 0}}.
%t M = {{0, 1, 0, 0, 0, 0},
%t {0, 0, 1, 0, 0, 0},
%t {0, 0, 0, 1, 0, 0},
%t {0, 0, 0, 0, 1, 0},
%t {0, 0, 0, 0, 0, 1},
%t {-1, 1, 0, 0, 1, 0}};
%t v[0] = Table[If[n == 6, 1, 0], {n, 1, 6}]
%t v[n_] := v[n] = M.v[n - 1];
%t b = Table[v[n][[1]], {n, 0, 50}]
%t CoefficientList[Series[x^5/((1-x)(1+x-x^5)),{x,0,60}],x] (* _Harvey P. Dale_, May 15 2021 *)
%o (Magma) M:=Matrix([[0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1], [-1, 1, 0, 0, 1, 0]]); V:=Matrix([[0],[0],[0],[0],[0],[1]]); [ (M^n*V)[1][1]: n in [0..60] ]; // _Klaus Brockhaus_, Nov 30 2010
%Y Cf. A174522.
%K sign,easy,less
%O 0,13
%A _Roger L. Bagula_, Nov 28 2010
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