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

0,1

Links

Table of n, a(n) for n=0..17.

Tanya Khovanova, Recursive Sequences.

Index entries for linear recurrences with constant coefficients, signature (29,1).

Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2).

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, ... .

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.

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

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

Sum_{n>=0} (-1)^n/(a(n)*a(n+1)) = 1/(26*sqrt(5)). - Amiram Eldar, Apr 06 2026

Example

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.

Mathematica

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

Prog

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

Crossrefs

Cf. A000032.

Sequence in context: A282735 A245252 A090251 * A380724 A389722 A024234

Adjacent sequences: A087278 A087279 A087280 * A087282 A087283 A087284

Keyword

easy,nonn

Author

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

Extensions

More terms from Ray Chandler, Feb 14 2004

More terms from Vincenzo Librandi, Apr 14 2011

Status

approved