A095003 a(n) = 9*a(n-1) - 9*a(n-2) + a(n-3).

1, 6, 45, 352, 2769, 21798, 171613, 1351104, 10637217, 83746630, 659335821, 5190939936, 40868183665, 321754529382, 2533168051389, 19943589881728, 157015551002433, 1236180818137734, 9732430994099437, 76623267134657760, 603253706083162641, 4749406381530643366

Offset: 1

Author

Gary W. Adamson, May 27 2004

Keywords

• nonn

Comments

a(n)/a(n-1) tends to 7.87298... = 4 + sqrt(15) = C (having the property that C + 1/C = 8). Eigenvalues of M are C, 1/C, 1; being roots of x^3 - 9x^2 + 9x - 1.

Examples

			a(4) = 352 since M^4 * [1 0 0] = [145, 352, 640].
		

Links

• Index entries for linear recurrences with constant coefficients, signature (9,-9,1).

Crossrefs

Cf. A095002, A095004, A076765.

Programs

Maple

a:= n-> (<<1|1|1>, <1|2|3>, <1|3|6>>^n)[1, 2]:
seq(a(n), n=1..23);  # Alois P. Heinz, Jun 06 2021

Mathematica

a[n_] := (MatrixPower[{{1, 1, 1}, {1, 2, 3}, {1, 3, 6}}, n].{{1}, {0}, {0}})[[2, 1]]; Table[ a[n], {n, 20}]; (* Robert G. Wilson v, May 29 2004 *)
LinearRecurrence[{9,-9,1},{1,6,45},30] (* Harvey P. Dale, Nov 12 2022 *)

Formula

a(n+3) = 9*a(n+2) - 9*a(n+1) + a(n); given a(1) = 1, a(2) = 6, a(3) = 45.

Let M be the 3 X 3 matrix [1 1 1 / 1 2 3 / 1 3 6]. M^n * [1 0 0] = [A095002(n) a(n) A095004(n)].

Extensions

Edited and extended by Robert G. Wilson v, May 29 2004

Definition corrected and edited by Georg Fischer, Jun 06 2021