%I #40 Jan 01 2024 11:07:23

%S 1,4,22,145,988,6766,46369,317812,2178310,14930353,102334156,

%T 701408734,4807526977,32951280100,225851433718,1548008755921,

%U 10610209857724,72723460248142,498454011879265,3416454622906708,23416728348467686

%N a(n) = Fibonacci(4n) + 1, or Fibonacci(2n-1)*Lucas(2n+1).

%C If x=a(n), y=a(n+1), z=a(n+2) or x=a(n+2), y=a(n+1), z=a(n), then x^2 -9*y*x +7*x*z +9*y^2 -9*z*y +z^2 = -45. - _Alexander Samokrutov_, Jul 02 2015

%D Hugh C. Williams, Edouard Lucas and Primality Testing, John Wiley and Sons, 1998, p. 75.

%H Colin Barker, <a href="/A081002/b081002.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (8,-8,1).

%F a(n) = 8a(n-1) - 8a(n-2) + a(n-3).

%F From _Paul Barry_, Jan 20 2005: (Start)

%F G.f.: (1-4*x-2*x^2)/(1-8*x+8*x^2-x^3).

%F a(n) = Sum_{k=0..n} binomial(2n-k, 2k)*2^(2n-3k).

%F a(n) = 1 - 0^n + Sum_{k=0..2n} binomial(4n-k-1, k). (End)

%p with(combinat): for n from 0 to 30 do printf(`%d,`,fibonacci(4*n)+1) od: # _James A. Sellers_, Mar 01 2003

%t Fibonacci[4*Range[0,30]]+1 (* or *) LinearRecurrence[{8,-8,1}, {1,4,22}, 30] (* _Harvey P. Dale_, Feb 26 2015 *)

%o (Magma) [Fibonacci(4*n)+1: n in [0..30]]; // _Vincenzo Librandi_, Apr 20 2011

%o (PARI) Vec((1-4*x-2*x^2)/((1-x)*(1-7*x+x^2)) + O(x^30)) \\ _Colin Barker_, Dec 23 2014

%o (Sage) [fibonacci(4*n)+1 for n in (0..30)] # _G. C. Greubel_, Jul 15 2019

%o (GAP) List([0..30], n-> Fibonacci(4*n)+1); # _G. C. Greubel_, Jul 15 2019

%Y Cf. A000045 (Fibonacci numbers), A000032 (Lucas numbers).

%K nonn,easy

%O 0,2

%A _R. K. Guy_, Mar 01 2003

%E More terms from _James A. Sellers_, Mar 01 2003