OFFSET
0,1
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.
LINKS
Indranil Ghosh, Table of n, a(n) for n = 0..530
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (76,1).
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
EXAMPLE
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.
MATHEMATICA
LucasL[9*Range[0, 20]] (* Paolo Xausa, Mar 04 2024 *)
PROG
(Magma) [ Lucas(9*n) : n in [0..100]]; // Vincenzo Librandi, Apr 14 2011
(PARI) a(n)=fibonacci(9*n-1)+fibonacci(9*n+1) \\ Charles R Greathouse IV, Feb 06 2017
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Oct 19 2003
EXTENSIONS
More terms from Vincenzo Librandi, Apr 14 2011
STATUS
approved