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%I #46 Sep 08 2022 08:45:32
%S 1,13,233,4181,75025,1346269,24157817,433494437,7778742049,
%T 139583862445,2504730781961,44945570212853,806515533049393,
%U 14472334024676221,259695496911122585,4660046610375530309,83621143489848422977,1500520536206896083277,26925748508234281076009
%N a(n) = Fibonacci(6*n+1).
%C For positive n, a(n) equals (-1)^n times the permanent of the (6n)X(6n) tridiagonal matrix with ((-1)^(1/6))'s along the three central diagonals. - _John M. Campbell_, Jul 12 2011
%C a(n) = x + y where those two values are solutions to: x^2 = 5*y^2 + 1. (See related sequences with formula below). - _Richard R. Forberg_, Sep 05 2013
%H Colin Barker, <a href="/A134493/b134493.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 From _R. J. Mathar_, Apr 17 2011: (Start)
%F G.f.: ( 1-5*x ) / ( 1-18*x+x^2 ).
%F a(n) = A049660(n+1) - 5*A049660(n). (End)
%F a(n) = Fibonacci(3*n+1)^2 + Fibonacci(3*n)^2. - _Gary Detlefs_, Oct 12 2011
%F a(n) = 18*a(n-1) - a(n-2). - _Richard R. Forberg_, Sep 05 2013
%F a(n) = A060645(n) + A023039(n), as derives from comment above. - _Richard R. Forberg_, Sep 05 2013
%F a(n) = ((5-sqrt(5)+(5+sqrt(5))*(9+4*sqrt(5))^(2*n)))/(10*(9+4*sqrt(5))^n). - _Colin Barker_, Jan 24 2016
%F 2*a(n) = Fibonacci(6*n) + Lucas(6*n). - _Bruno Berselli_, Oct 13 2017
%F a(n) = S(n, 18) - 5*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
%t Table[Fibonacci[6n+1], {n, 0, 30}]
%o (Magma) [Fibonacci(6*n+1): n in [0..100]]; // _Vincenzo Librandi_, Apr 16 2011
%o (PARI) a(n)=fibonacci(6*n+1) \\ _Charles R Greathouse IV_, Jul 15 2011
%o (PARI) Vec((1-5*x)/(1-18*x+x^2) + O(x^100)) \\ _Altug Alkan_, Jan 24 2016
%Y Cf. A000032, A000045, A049660, A134492, A134494, A134495, A103134, A134497.
%K nonn,easy
%O 0,2
%A _Artur Jasinski_, Oct 28 2007
%E Offset changed to 0 by _Vincenzo Librandi_, Apr 16 2011
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