A095002 a(n) = 9*a(n-1) - 9*a(n-2) + a(n-3); given a(1) = 1, a(2) = 3, a(3) = 19.

1, 3, 19, 145, 1137, 8947, 70435, 554529, 4365793, 34371811, 270608691, 2130497713, 16773373009, 132056486355, 1039678517827, 8185371656257, 64443294732225, 507360986201539, 3994444594880083, 31448195772839121, 247591121587832881, 1949280776929823923

Offset: 1

Author

Gary W. Adamson, May 27 2004

Keywords

• nonn
• easy

Comments

A companion to A095003, A005004; a(n)/a(n-1) tending to 4 + sqrt(15).

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

Examples

			a(4) = 145 = 9*19 - 9*3 + 1.
a(4) = 145, leftmost term in M^4 * [1 0 0] = [145 352 640].
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (9,-9,1).

Crossrefs

Cf. A095003, 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}})[[1, 1]]; Table[ a[n], {n, 20}]; (* Robert G. Wilson v, May 29 2004 *)
nxt[{a_,b_,c_}]:={b,c,9c-9b+a}; NestList[nxt,{1,3,19},30][[All,1]] (* Harvey P. Dale, Sep 02 2022 *)

PARI

Vec(x*(1-6*x+x^2)/((1-x)*(1-8*x+x^2)) + O(x^20)) \\ Michel Marcus, Mar 21 2015

Formula

Let M be the 3 X 3 matrix [1 1 1 / 1 2 3 / 1 3 6]. M^n * [1 0 0] = [a(n) A095003(n) A095004(n)].

From R. J. Mathar, Aug 22 2008: (Start)

O.g.f.: x*(1-6x+x^2)/((1-x)*(1-8x+x^2)).

a(n) = (2 + A001090(n+1) - 7*A001090(n))/3. (End)

Extensions

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

Edited by Georg Fischer, Jun 06 2021