%I #42 Aug 13 2022 05:15:47

%S 5,89,1597,28657,514229,9227465,165580141,2971215073,53316291173,

%T 956722026041,17167680177565,308061521170129,5527939700884757,

%U 99194853094755497,1779979416004714189,31940434634990099905,573147844013817084101,10284720757613717413913

%N a(n) = Fibonacci(6n+5).

%H Colin Barker, <a href="/A134497/b134497.txt">Table of n, a(n) for n = 0..750</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (18,-1).

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%F G.f.: ( 5-x ) / ( 1-18*x+x^2 ). a(n) = 5*A049660(n+1)-A049660(n). - _R. J. Mathar_, Apr 17 2011

%F a(n) = A000045(A016969(n)). - _Michel Marcus_, Nov 08 2013

%F 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

%F 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

%F From _Peter Bala_, Aug 11 2022: (Start)

%F 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).

%F 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)

%t Table[Fibonacci[6n+5], {n, 0, 30}]

%t Take[Fibonacci[Range[100]],{5,-1,6}] (* _Harvey P. Dale_, Jun 18 2013 *)

%o (Magma) [Fibonacci(6*n +5): n in [0..100]]; // _Vincenzo Librandi_, Apr 17 2011

%o (PARI) a(n)=fibonacci(6*n+5) \\ _Charles R Greathouse IV_, Jun 11 2015

%o (PARI) Vec((5-x)/(1-18*x+x^2) + O(x^100)) \\ _Altug Alkan_, Jan 24 2016

%Y Cf. A000045, A049660, A101858, A103134, A134492, A134493, A134494, A134495.

%K nonn,easy

%O 0,1

%A _Artur Jasinski_, Oct 28 2007

%E Offset changed from 1 to 0 by _Vincenzo Librandi_, Apr 17 2011