A095004 a(n) = 9*a(n-1) - 9*a(n-2) + a(n-3); given a(1) = 1, a(2) = 10, a(3) = 81.

1, 10, 81, 640, 5041, 39690, 312481, 2460160, 19368801, 152490250, 1200553201, 9451935360, 74414929681, 585867502090, 4612525087041, 36314333194240, 285902140466881, 2250902790540810, 17721320183859601, 139519658680336000, 1098435949258828401, 8647967935390291210

Offset

1,2

Comments

A sequence derived from A076765, with a(n)/a(n-1) tending to 4 + sqrt(15).

a(n)/a(n-1) tends to C = 4 + sqrt(15) = 7.87298334... (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.

This is the r=10 member of the r-family of sequences S_r(n), n>=1, defined in A092184, where more information can be found.

Links

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

Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, Polynomial sequences on quadratic curves, Integers, Vol. 15, 2015, #A38.

Index entries for sequences related to Chebyshev polynomials.

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

Formula

a(n) = A076765(n-1) + A076765(n-2).

Let M be the 3 X 3 matrix [1 1 1 / 1 2 3 / 1 3 6]; then M^n * [1 0 0] = [A095002(n) A095003(n) a(n)].

a(n)= (T(n, 4)-1)/3 with Chebyshev's polynomials of the first kind evaluated at x=4: T(n, 4)=A001091(n). a(0):=0. - Wolfdieter Lang, Oct 18 2004

G.f.: x*(1+x)/((1-x)*(1-8*x+x^2)) = x*(1+x)/(1-9*x+9*x^2-x^3).

Example

a(4) = 640 = 568 + 72 = A076765(3) + A076765(2).
a(4) = 640 = 9*81 - 9*10 + 1.
a(4) = 640, rightmost term in M^4 * [1 0 0]: [145 352 640] = [A095002(4) A095003(4) A095004(4)].

Maple

a:= n-> (<<1|1|1>, <1|2|3>, <1|3|6>>^n)[1, 3]:
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}})[[3, 1]]; Table[ a[n], {n, 20}]; (* Robert G. Wilson v, May 29 2004 *)

Crossrefs

Cf. A095002, A095003, A076765.

Sequence in context: A293367 A018202 A154236 * A037541 A037485 A343276

Adjacent sequences: A095001 A095002 A095003 * A095005 A095006 A095007

Keyword

nonn,easy

Author

Gary W. Adamson, May 27 2004

Extensions

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

Definition aligned with A095002, A095003 by Georg Fischer, Jun 06 2021

Status

approved