A136270 a(n) = 20*a(n-1) - 3*a(n-2).

1, 17, 337, 6689, 132769, 2635313, 52307953, 1038253121, 20608138561, 409048011857, 8119135821457, 161155572393569, 3198754040407009, 63491614090959473, 1260236019697968433, 25014245551686490241

Offset: 1

Author

Gary W. Adamson, Mar 19 2008

Keywords

• nonn
• easy

Comments

a(n)/a(n-1) tends to (sqrt(97) + 10), an eigenvalue of the matrix and root of the characteristic polynomial x^2 - 20x + 3.

Examples

			a(4) = 20*a(3) - 3*a(2) = 20*337 - 3*17.
[a(3), a(4)] = [0,1; -3,20] ^3 * [1,1] = [337, 6689].
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..765
• Index entries for linear recurrences with constant coefficients, signature (20,-3).

Crossrefs

Cf. A137246.

Programs

Mathematica

LinearRecurrence[{20, -3}, {1, 17}, 50] (* G. C. Greubel, Feb 23 2017 *)

PARI

x='x+O('x^50); Vec((1-3*x)/(1-20*x+3*x^2)) \\ G. C. Greubel, Feb 23 2017

Formula

a(n) = 20*a(n-1) - 3*a(n-2), n>2; a(1) = 1, a(2) = 17.

[a(3), a(4)] = the 2 X 2 matrix [0,1; -3,20]^n * [1,1].

A137246(n) = 20*a(n) - 3*a(n-1), n>4.

O.g.f.: (1-3*x)/(1-20*x+3*x^2). - R. J. Mathar and Alexander R. Povolotsky, Mar 31 2008

a(n) = (1/2)*(10 - sqrt(97))^n - (9/194)*sqrt(97)*(10 + sqrt(97))^n + (1/2)*(10 + sqrt(97))^n + (9/194)*(10 - sqrt(97))^n*sqrt(97) - Alexander R. Povolotsky, Mar 31 2008