A122788 (1,3)-entry of the 3 X 3 matrix M^n, where M = {{0, -1, 1}, {1, 1, 0}, {0, 1, 1}}.

0, 1, 1, 0, 0, 2, 4, 4, 4, 8, 16, 24, 32, 48, 80, 128, 192, 288, 448, 704, 1088, 1664, 2560, 3968, 6144, 9472, 14592, 22528, 34816, 53760, 82944, 128000, 197632, 305152, 471040, 727040, 1122304, 1732608, 2674688, 4128768, 6373376, 9838592, 15187968, 23445504

Offset: 0

Author

Gary W. Adamson and Roger L. Bagula, Oct 20 2006

Keywords

• nonn
• easy

Comments

Essentially the same as A078003: a(n) = A078003(n-1).

Examples

			a(7)=4 because M^7 = {{0,4,4},{4,4,8},{8,12,12}}.
		

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,-2,2).

Programs

Maple

with(linalg): M[1]:=matrix(3,3,[0,-1,1,1,1,0,0,1,1]): for n from 2 to 42 do M[n]:=multiply(M[1],M[n-1]) od: 0,seq(M[n][1,3],n=1..42);
a[0]:=0: a[1]:=1: a[2]:=1: for n from 3 to 42 do a[n]:=2*a[n-1]-2*a[n-2]+2*a[n-3] od: seq(a[n],n=0..42);

Mathematica

M = {{0, -1, 1}, {1, 1, 0}, {0, 1, 1}}; v[1] = {0, 0, 1}; v[n_] := v[n] = M.v[n - 1]; a1 = Table[v[n][[1]], {n, 1, 50}]

PARI

concat(0, Vec(x*(1 - x) / (1 - 2*x + 2*x^2 - 2*x^3) + O(x^50))) \\ Colin Barker, Mar 03 2017

Formula

Recurrence relation a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) (follows from the minimal polynomial of the matrix M).

a(n) = A078003(n-1). - R. J. Mathar, Aug 02 2008

G.f.: x*(1 - x) / (1 - 2*x + 2*x^2 - 2*x^3). - Colin Barker, Mar 03 2017

Extensions

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