A123189 a(0) = 0, a(1) = 1, a(2)=1; a(n) = 3a(n-1) + 3a(n-2) - 4a(n-3) for n >= 3.

0, 1, 1, 6, 17, 65, 222, 793, 2785, 9846, 34721, 122561, 432462, 1526185, 5385697, 19005798, 67069745, 236683841, 835237566, 2947485241, 10401433057, 36705804630, 129531772097, 457106997953, 1613093091630, 5692473180361

Offset

0,4

Comments

a(n) is the sum of the upper-left and the upper-middle element of {{3,3,-4},{1,0,0},{0,1,0}}^(n-2). - Jianing Song, Aug 11 2023

Links

Jianing Song, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,3,-4).

Formula

G.f.: (x-2*x^2)/(1-3*x-3*x^2+4*x^3). - Jianing Song, Aug 11 2023

Maple

a[1]:=1: a[2]:=1: a[3]:=6: for n from 4 to 25 do a[n]:=3*a[n-1]+3*a[n-2]-4*a[n-3] od: seq(a[n], n=1..25);

Mathematica

M = {{1, -2, 1}, {-2, 2, 0}, {1, 1, 0}} v[1] = {1, 0, 0} v[n_] := v[n] = M.v[n - 1]; a1 = Table[v[n][[1]], {n, 1, 50}]

Prog

(PARI) a(n) = my(Mn=[3, 3, -4; 1, 0, 0; 0, 1, 0]^(n-2)); Mn[1, 1]+Mn[1, 2] \\ Jianing Song, Aug 11 2023

Crossrefs

Sequence in context: A195741 A006758 A363165 * A219847 A047156 A128243

Adjacent sequences: A123186 A123187 A123188 * A123190 A123191 A123192

Keyword

nonn

Author

Roger L. Bagula, Oct 03 2006

Extensions

Edited by N. J. A. Sloane, Oct 08 2006

Added initial a(0) = 0 at the suggestion of Jianing Song. - N. J. A. Sloane, Aug 12 2023

Status

approved