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

1,2

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.

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).

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)

Example

a(4) = 145 = 9*19 - 9*3 + 1.
a(4) = 145, leftmost term in 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}})[[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 *)

Prog

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

Crossrefs

Cf. A095003, A095004, A076765.

Sequence in context: A058859 A291964 A333094 * A293527 A080833 A073516

Adjacent sequences: A094999 A095000 A095001 * A095003 A095004 A095005

Keyword

nonn,easy

Author

Gary W. Adamson, May 27 2004

Extensions

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

Edited by Georg Fischer, Jun 06 2021

Status

approved