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%I #9 Aug 09 2015 01:22:44
%S 1,-1,1,-1,1,0,4,10,33,93,264,729,1999,5437,14726,39757,107118,288201,
%T 774672,2080936,5587388,14997840,40249449,108001720,289774494,
%U 777430359,2085660586,5595162045,15009725921,40264911876,108013136968,289750079191,777264097335,2085030095791
%N First row sum of the 5 X 5 matrix M^n, where M = {{0, -1, 0, 0, 0}, {-1, 1, -1, 0, 0}, {0, -1, 1, -1, 0}, {0, 0, -1, 1, -1}, {0, 0, 0, -1, 1}}, n>=0.
%F a(n)=4a(n-1)-2a(n-2)-5a(n-3)+2a(n-4)+a(n-5) (follows from the minimal polynomial of the matrix M).
%F G.f.:(2*x-1)*(x^2-3*x+1)/(-1+x^5+2*x^4-5*x^3-2*x^2+4*x). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009
%e a(5)=0 because first row of M^5 is {6,-15,15,-10,4}.
%p with(linalg): M[1]:=matrix(5,5,[0,-1,0,0,0,-1,1,-1,0,0,0,-1,1,-1,0,0,0,-1,1,-1,0,0,0,-1,1]): for n from 2 to 33 do M[n]:=multiply(M[1],M[n-1]) od: 1,seq(add(M[n][1,j],j=1..5),n=1..33);
%p a[0]:=1: a[1]:=-1: a[2]:=1: a[3]:=-1: a[4]:=1: for n from 5 to 33 do a[n]:=4*a[n-1]-2*a[n-2]-5*a[n-3]+2*a[n-4]+a[n-5] od: seq(a[n],n=0..33);
%t M = {{0, -1, 0, 0, 0}, {-1, 1, -1, 0, 0}, {0, -1, 1, -1, 0}, {0, 0, -1, 1, -1}, {0, 0, 0, -1, 1}}; v[1] = {1, 1, 1, 1, 1}; v[n_] := v[n] = M.v[n - 1]; a1 = Table[v[n][[1]], {n, 1, 50}]
%Y Cf. A098493.
%K sign
%O 0,7
%A _Gary W. Adamson_ and _Roger L. Bagula_, Oct 24 2006
%E Edited by _N. J. A. Sloane_, Nov 24 2006
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