A087287 a(n) = Lucas(9*n).

2, 76, 5778, 439204, 33385282, 2537720636, 192900153618, 14662949395604, 1114577054219522, 84722519070079276, 6440026026380244498, 489526700523968661124, 37210469265847998489922, 2828485190904971853895196, 215002084978043708894524818, 16342986943522226847837781364, 1242282009792667284144565908482

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 (76 + sqrt(5780))/2 = 76.01315561749...

a(0)/a(1) = 2/76, a(1)/a(2) = 76/5778, a(2)/a(3) = 5778/439204, a(3)/a(4) = 439204/33385282, etc.

Lim_{n->oo} a(n)/a(n+1) = 0.01315561749... = 2/(76 + sqrt(5780)) = (sqrt(5780) - 76)/2.

Examples

			a(4) = 33385282 = 76*a(3) + a(2) = 76*439204 + 5778 = ((76 + sqrt(5780))/2)^4 + ((76 - sqrt(5780))/2)^4 = 33385281.999999970046... + 0.000000029953... = 33385282.
		

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..530
• 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 (76,1).

Crossrefs

Cf. A000032.

Programs

Magma

[ Lucas(9*n) : n in [0..100]]; // Vincenzo Librandi, Apr 14 2011

Mathematica

LucasL[9*Range[0, 20]] (* Paolo Xausa, Mar 04 2024 *)

PARI

a(n)=fibonacci(9*n-1)+fibonacci(9*n+1) \\ Charles R Greathouse IV, Feb 06 2017

Formula

a(n) = 76a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 76.

a(n) = ((76 + sqrt(5780))/2)^n + ((76 - sqrt(5780))/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.: (2-76*x)/(1-76*x-x^2). - Philippe Deléham, Nov 02 2008

E.g.f.: 2*exp(38*x)*cosh(17*sqrt(5)*x). - Stefano Spezia, Jan 18 2025

Extensions

More terms from Vincenzo Librandi, Apr 14 2011