A087281 a(n) = Lucas(7*n).

2, 29, 843, 24476, 710647, 20633239, 599074578, 17393796001, 505019158607, 14662949395604, 425730551631123, 12360848946698171, 358890350005878082, 10420180999117162549, 302544139324403592003, 8784200221406821330636, 255044350560122222180447, 7405070366464951264563599

Offset: 0

Author

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Oct 19 2003

Keywords

• easy
• nonn

Comments

a(n+1)/a(n) converges to (29+sqrt(845))/2 = 29.0344418537...

a(0)/a(1) = 2/29, a(1)/a(2) = 29/843, a(2)/a(3) = 843/24476, a(3)/a(4) = 24476/710647, etc.

Lim_{n->oo} a(n)/a(n+1) = 0.0344418537... = 2/(29+sqrt(845)) = (sqrt(845)-29)/2.

Examples

			a(4) = 710647 = 29*a(3) + a(2) = 29*24476 + 843 = ((29+sqrt(845))/2)^4 + ((29-sqrt(845))/2)^4 = 710646.9999985928... + 0.0000014071... = 710647.
		

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 (29,1).

Crossrefs

Cf. A000032.

Programs

Magma

[ Lucas(7*n) : n in [0..100]]; // Vincenzo Librandi, Apr 14 2011

Mathematica

LucasL[7Range[0,20]] (* or *) LinearRecurrence[{29,1},{2,29},20] (* Harvey P. Dale, Nov 22 2011 *)

Formula

a(n) = 29*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 29.

a(n) = ((29 + sqrt(845))/2)^n + ((29 - sqrt(845))/2)^n.

a(n)^2 = a(2n) - 2 for n = 1, 3, 5, ...;

a(n)^2 = a(2n) + 2 for n = 2, 4, 6, ....

G.f.: (2-29*x)/(1-29*x-x^2). - Philippe Deléham, Nov 02 2008

E.g.f.: 2*exp(29*x/2)*cosh(13*sqrt(5)*x/2). - Stefano Spezia, Jan 18 2025

Extensions

More terms from Ray Chandler, Feb 14 2004

More terms from Vincenzo Librandi, Apr 14 2011