1,3

M = {{0, 1, 1, -1, -1}, {1, 0, 1, -1, -1}, {1, 1, 0, -1, -1}, {-1, -1, -1, 2, 1}, {-1, -1, -1, 1, 2}}; w[1] = {1, 0, 0, 0, 0}; w[n_] := w[n] = M.w[n - 1] a(n) =w[n][[1]]

O.g.f.: (x^2+4*x-1)/((x+1)*(5*x-1)). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Dec 05 2007

a(n)=the (1, 1)-term of M^(n-1), where M=matrix(5, 5, [0, 1, 1, -1, -1; 1, 0, 1, -1, -1; 1, 1, 0, -1, -1; -1, -1, -1, 2, 1; -1, -1, -1, 1, 2]) (the second Maple program makes use of this formula). The minimal polynomial of M is x^4-5x^3-x^2+5x, the coefficients of which yield the coefficients of the recurrence relation.

a[1]:=1: a[2]:=0: a[3]:=4: a[4]:=16: for n from 5 to 25 do a[n]:=5*a[n-1]+a[n-2]-5*a[n-3] od: seq(a[n], n=1..25);

with(linalg): M[1]:=matrix(5, 5, [0, 1, 1, -1, -1, 1, 0, 1, -1, -1, 1, 1, 0, -1, -1, -1, -1, -1, 2, 1, -1, -1, -1, 1, 2]): for n from 2 to 24 do M[n]:=multiply(M[1], M[n-1]) od: 1, seq(M[n][1, 1], n=1..24);

M = {{0, 1, 1, -1, -1}, {1, 0, 1, -1, -1}, {1, 1, 0, -1, -1}, {-1, -1, -1, 2, 1}, {-1, -1, -1, 1, 2}}; w[1] = {1, 0, 0, 0, 0}; w[n_] := w[n] = M.w[n - 1] a = Table[w[n][[1]], {n, 1, 30}]

Sequence in context: A076997 A092304 A022564 this_sequence A121146 A134006 A090013

Adjacent sequences: A123007 A123008 A123009 this_sequence A123011 A123012 A123013

nonn

Roger Bagula (rlbagulatftn(AT)yahoo.com), Sep 23 2006

Edited by N. J. A. Sloane (njas(AT)research.att.com), Oct 15 2006