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%I #15 Sep 21 2017 03:24:38
%S 1,30,707,16886,403104,9623140,229729153,5484227157,130922641160,
%T 3125460977225,74612811302754,1781200165693270,42521840081752984,
%U 1015105948653689061,24233196047277585233,578508865448619225434
%N First row sum of the 4 X 4 matrix M^n, where M={{10, 9, 7, 4}, {9, 8, 6, 3}, {7, 6, 4, 2}, {4, 3, 2, 1}}.
%C These matrices resemble Hankel matrices.
%H P. Steinbach, <a href="http://www.jstor.org/stable/2691048">Golden fields: a case for the heptagon</a>, Math. Mag. 70 (1997), no. 1, 22-31.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (23,21,-4,-1).
%F a(n) = 23a(n-1)+21a(n-2)-4a(n-3)-a(n-4) for n>=4; a(0)=1, a(1)=30, a(2)=707, a(3)=16886 (follows from the minimal polynomial x^4-23x^3-21x^2+4x+1 of the matrix M).
%F G.f.: -(x^3+4*x^2-7*x-1) / ((x+1)*(x^3+3*x^2-24*x+1)). [_Colin Barker_, Dec 07 2012]
%p with(linalg): M[1]:=matrix(4,4,[10,9,7,4,9,8,6,3,7,6,4,2,4,3,2,1]): for n from 2 to 15 do M[n]:=multiply(M[1],M[n-1]) od: 1,seq(M[n][1,1]+M[n][1,2]+M[n][1,3]+M[n][1,4],n=1..15);
%p a[0]:=1: a[1]:=30: a[2]:=707: a[3]:=16886: for n from 4 to 15 do a[n]:=23*a[n-1]+21*a[n-2]-4*a[n-3]-a[n-4] od: seq(a[n],n=0..15);
%t M = {{10, 9, 7, 4}, {9, 8, 6, 3}, {7, 6, 4, 2}, {4, 3, 2, 1}}; v[1] = {1, 1, 1, 1}; v[n_] := v[n] = M.v[n - 1]; a1 = Table[v[n][[1]], {n, 1, 50}]
%Y Cf. A015448.
%K nonn,easy
%O 0,2
%A _Gary W. Adamson_ and _Roger L. Bagula_, Oct 17 2006
%E Edited by _N. J. A. Sloane_, Nov 07 2006
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