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

1,2

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.

Links

Table of n, a(n) for n=1..22.

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

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)].

Example

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

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 *)

Crossrefs

Cf. A095002, A095004, A076765.

Sequence in context: A024077 A097129 A048663 * A004988 A257633 A371779

Adjacent sequences: A095000 A095001 A095002 * A095004 A095005 A095006

Keyword

nonn

Author

Gary W. Adamson, May 27 2004

Extensions

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

Definition corrected and edited by Georg Fischer, Jun 06 2021

Status

approved