A086928 a(n) = 12*a(n-1) + a(n-2), with a(0) = 2 and a(1) = 12.

2, 12, 146, 1764, 21314, 257532, 3111698, 37597908, 454286594, 5489037036, 66322731026, 801361809348, 9682664443202, 116993335127772, 1413602685976466, 17080225566845364, 206376309488120834

Offset: 0

Author

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Sep 21 2003

Keywords

• easy
• nonn

Comments

a(n+1)/a(n) converges to (6+sqrt(37)) = 12.0827625... a(0)/a(1)=2/12; a(1)/a(2)=12/146; a(2)/a(3)=146/1764; a(3)/a(4)=1764/21314; ... etc.

Lim_{n->infinity} a(n)/a(n+1) = 0.0827625... = 1/(6+sqrt(37)) = sqrt(37) - 6.

Examples

			a(4) = 21314 = 12*a(3) + a(2) = 12*1764 + 146 = (6+sqrt(37))^4 + (6-sqrt(37))^4 = 21313.999953 + 0.000047 = 21314.
		

Links

• 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. A001927.

Programs

Mathematica

LinearRecurrence[{12,1},{2,12},20] (* Harvey P. Dale, Oct 31 2016 *)

Formula

a(n) = (6+sqrt(37))^n + (6-sqrt(37))^n.

G.f.: (2-12*x)/(1-12*x-x^2). - Philippe Deléham, Nov 21 2008