A087800 a(n) = 12*a(n-1) - a(n-2), with a(0) = 2 and a(1) = 12.

2, 12, 142, 1692, 20162, 240252, 2862862, 34114092, 406506242, 4843960812, 57721023502, 687808321212, 8195978831042, 97663937651292, 1163771272984462, 13867591338162252, 165247324784962562, 1969100306081388492

Offset: 0

Author

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Oct 11 2003

Keywords

• easy
• nonn

Comments

a(n+1)/a(n) converges to (6+sqrt(35)) = 11.9160797... a(0)/a(1)=2/12; a(1)/a(2)=12/142; a(2)/a(3)=142/1692; a(3)/a(4)=1692/20162; ... etc. Lim_{n->infinity} a(n)/a(n+1) = 0.0839202... = 1/(6+sqrt(35)) = (6-sqrt(35)).

Except for the first term, positive values of x (or y) satisfying x^2 - 12xy + y^2 + 140 = 0. - Colin Barker, Feb 25 2014

Examples

			a(4) = 20162 = 12a(3) - a(2) = 12*1692 - 142 = (6+sqrt(35))^4 + (6-sqrt(35))^4 = 20161.9999504 + 0.00004959 = 20162.
G.f. = 2 + 12*x + 142*x^2 + 1692*x^3 + 20162*x^4 + 240252*x^5 + ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Tanya Khovanova, Recursive Sequences
• Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
• Index entries for linear recurrences with constant coefficients, signature (12,-1).

Crossrefs

Cf. A009747, A086928, A001927, A023038.

Programs

Magma

I:=[2,12]; [n le 2 select I[n] else 12*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Nov 07 2018

Mathematica

a[0] = 2; a[1] = 12; a[n_] := 12a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 17}] (* Robert G. Wilson v, Jan 30 2004 *)
CoefficientList[Series[(2 - 12 x)/(1 - 12 x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 28 2014 *)
a[ n_] := 2 ChebyshevT[ n, 6]; (* Michael Somos, May 28 2014 *)
LinearRecurrence[{12,-1},{2,12},20] (* Harvey P. Dale, Jan 29 2019 *)

PARI

Vec((2-12*x)/(1-12*x+x^2) + O(x^100)) \\ Colin Barker, Feb 25 2014

PARI

{a(n) = 2 * polchebyshev( n, 1, 6)}; /* Michael Somos, May 28 2014 */

Sage

[lucas_number2(n,12,1) for n in range(1,20)] # Zerinvary Lajos, Jun 25 2008

Formula

a(n) = (6+sqrt(35))^n + (6-sqrt(35))^n.

a(n) = 2*A023038(n).

G.f.: (2-12*x)/(1-12*x+x^2). - Philippe Deléham, Nov 17 2008

a(-n) = a(n). - Michael Somos, May 28 2014