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A122607 Expansion of x*(8*x^5 + 5*x^4 - x^3 - 5*x^2 - 1)/(x^6 + 3*x^5 + 6*x^4 + 4*x^3 - 5*x^2 + x - 1).

%I A122607 #22 Feb 17 2024 11:14:07
%S A122607 1,1,1,1,1,1,10,19,-17,-62,163,550,-548,-3050,2665,19450,-7550,
%T A122607 -113534,8308,667423,187462,-3800825,-2366747,21303154,21068938,
%U A122607 -116488961,-162036530,621601885,1153785034,-3216794309,-7799929064,16026195376,50784142789,-75764359214,-320876463932
%N A122607 Expansion of x*(8*x^5 + 5*x^4 - x^3 - 5*x^2 - 1)/(x^6 + 3*x^5 + 6*x^4 + 4*x^3 - 5*x^2 + x - 1).
%C A122607 Obtained as the top element of the vector resulting from multiplying the n-th power of the 6 X 6 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, 3, 6, 4, -5, 1]] with the column vector which contains only 1's.
%H A122607 Peter Steinbach, <a href="http://www.jstor.org/stable/2691048">Golden fields: a case for the heptagon</a>, Math. Mag. Vol. 70, No. 1, Feb. 1997, 22-31.
%H A122607 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,-5,4,6,3,1).
%F A122607 G.f.: x*(8*x^5+5*x^4-x^3-5*x^2-1)/(x^6+3*x^5+6*x^4+4*x^3-5*x^2+x-1). - _Colin Barker_, Nov 08 2012
%t A122607 M = {{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, 3, 6, 4, -5, 1}}; v[1] = {1, 1, 1, 1, 1, 1}; v[n_] := v[n] = M.v[n - 1] a = Table[Floor[v[n][[1]]], {n, 1, 50}]
%t A122607 LinearRecurrence[{1,-5,4,6,3,1},{1,1,1,1,1,1},40] (* _Harvey P. Dale_, Feb 17 2024 *)
%Y A122607 Cf. A066170.
%K A122607 sign,easy
%O A122607 1,7
%A A122607 _Roger L. Bagula_ and _Gary W. Adamson_, Sep 20 2006
%E A122607 Edited by _N. J. A. Sloane_, Sep 24 2006
%E A122607 Definition changed using Barker's g.f. by _Bruno Berselli_, Sep 19 2017