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A134497
a(n) = Fibonacci(6n+5).
12
5, 89, 1597, 28657, 514229, 9227465, 165580141, 2971215073, 53316291173, 956722026041, 17167680177565, 308061521170129, 5527939700884757, 99194853094755497, 1779979416004714189, 31940434634990099905, 573147844013817084101, 10284720757613717413913
OFFSET
0,1
FORMULA
G.f.: ( 5-x ) / ( 1-18*x+x^2 ). a(n) = 5*A049660(n+1)-A049660(n). - R. J. Mathar, Apr 17 2011
a(n) = A000045(A016969(n)). - Michel Marcus, Nov 08 2013
a(n) = ((25-11*sqrt(5)+(9+4*sqrt(5))^(2*n)*(25+11*sqrt(5))))/(10*(9+4*sqrt(5))^n). - Colin Barker, Jan 24 2016
a(n) = 5*S(n, 18) - S(n-1, 18), n >= 0, with the Chebyshev S-polynomials S(n-1, 18) = A049660(n). (See the g.f.) - Wolfdieter Lang, Jul 10 2018
From Peter Bala, Aug 11 2022: (Start)
Let n ** m = n*m + floor(phi*n)*floor(phi*m), where phi = (1 + sqrt(5))/2, denote the Porta-Stolarsky star product of the integers n and m (see A101858). Then a(n) = 5 ** 5 ** ... ** 5 (n+1 factors).
a(2*n+1) = a(n) ** a(n) = Fibonacci(12*n+11); a(3*n+2) = a(n) ** a(n) ** a(n) = Fibonacci(18*n+17) and so on. (End)
MATHEMATICA
Table[Fibonacci[6n+5], {n, 0, 30}]
Take[Fibonacci[Range[100]], {5, -1, 6}] (* Harvey P. Dale, Jun 18 2013 *)
PROG
(Magma) [Fibonacci(6*n +5): n in [0..100]]; // Vincenzo Librandi, Apr 17 2011
(PARI) a(n)=fibonacci(6*n+5) \\ Charles R Greathouse IV, Jun 11 2015
(PARI) Vec((5-x)/(1-18*x+x^2) + O(x^100)) \\ Altug Alkan, Jan 24 2016
KEYWORD
nonn,easy
AUTHOR
Artur Jasinski, Oct 28 2007
EXTENSIONS
Offset changed from 1 to 0 by Vincenzo Librandi, Apr 17 2011
STATUS
approved