A122174 First row sum of the matrix M^n, where M is the 5 X 5 matrix {{0,-1,-1,-1,-1}, {-1,0,-1,-1,0}, {-1,-1,0,0,0}, {-1,-1,0,1,0}, {-1,0,0,0,1}}.

1, -4, 6, -24, 41, -145, 273, -886, 1789, -5457, 11605, -33807, 74761, -210366, 479256, -1313465, 3061242, -8222492, 19501429, -51579259, 123983182, -324067194, 787044384, -2038584810, 4990387355, -12836179872, 31614557443, -80883958143, 200146505560, -509959672813

Offset

0,2

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Peter Steinbach, Golden fields: a case for the heptagon, Math. Mag. Vol. 70, No. 1, Feb. 1997, 22-31.

Index entries for linear recurrences with constant coefficients, signature (2,5,-13,7,-1).

Formula

a(n) = 2*a(n-1)+5*a(n-2)-13*a(n-3)+7*a(n-4)-a(n-5); a(0)=1, a(1)=-4, a(2)=6, a(3)=-24, a(4)=41 (follows from the minimal polynomial x^5-2*x^4-5*x^3+13*x^2-7*x+1 of the matrix M).

G.f.: (1-3*x^3+9*x^2-6*x)/(1+x^5-7*x^4+13*x^3-5*x^2-2*x). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009

Maple

with(linalg): M[1]:=matrix(5, 5, [0, -1, -1, -1, -1, -1, 0, -1, -1, 0, -1, -1, 0, 0, 0, -1, -1, 0, 1, 0, -1, 0, 0, 0, 1]): for n from 2 to 30 do M[n]:=multiply(M[n-1], M[1]) od: 1, seq(M[n][1, 1]+M[n][1, 2]+M[n][1, 3]+M[n][1, 4]+M[n][1, 5], n=1..30);

Mathematica

M = {{0, -1, -1, -1, -1}, {-1, 0, -1, -1, 0}, {-1, -1, 0, 0, 0}, {-1, -1, 0, 1, 0}, {-1, 0, 0, 0, 1}}; v[1] = {1, 1, 1, 1, 1}; v[n_] := v[n] = M.v[n - 1]; a1 = Table[v[n][[1]], {n, 1, 25}]

Prog

(PARI) a(n) = my(m=[0, -1, -1, -1, -1; -1, 0, -1, -1, 0; -1, -1, 0, 0, 0; -1, -1, 0, 1, 0; -1, 0, 0, 0, 1]); vecsum((m^n)[1, ]); \\ Michel Marcus, Jun 21 2017

Crossrefs

Cf. A046854, A007700, A059455, A065941.

Sequence in context: A164532 A098660 A363631 * A283185 A034458 A240290

Adjacent sequences: A122171 A122172 A122173 * A122175 A122176 A122177

Keyword

sign,easy,less

Author

Gary W. Adamson and Roger L. Bagula, Oct 17 2006

Extensions

Edited by N. J. A. Sloane, Oct 29 2006

Status

approved