A122948 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.

1, -1, 1, -1, 1, 0, 4, 10, 33, 93, 264, 729, 1999, 5437, 14726, 39757, 107118, 288201, 774672, 2080936, 5587388, 14997840, 40249449, 108001720, 289774494, 777430359, 2085660586, 5595162045, 15009725921, 40264911876, 108013136968, 289750079191, 777264097335, 2085030095791

Offset

0,7

Links

Table of n, a(n) for n=0..33.

Formula

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).

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

Example

a(5)=0 because first row of M^5 is {6,-15,15,-10,4}.

Maple

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);
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);

Mathematica

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}]

Crossrefs

Cf. A098493.

Sequence in context: A052372 A052373 A007716 * A317800 A357799 A149171

Adjacent sequences: A122945 A122946 A122947 * A122949 A122950 A122951

Keyword

sign

Author

Gary W. Adamson and Roger L. Bagula, Oct 24 2006

Extensions

Edited by N. J. A. Sloane, Nov 24 2006

Status

approved