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

Author

Gary W. Adamson, May 27 2004

Keywords

• nonn
• easy

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.

Examples

			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)].
		

Links

• 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).

Crossrefs

Cf. A095002, A095003, A076765.

Programs

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 *)

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).

Extensions

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

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