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%I #34 Dec 16 2023 17:45:12
%S 5,30,200,1365,9350,64080,439205,3010350,20633240,141422325,969323030,
%T 6643838880,45537549125,312119004990,2139295485800,14662949395605,
%U 100501350283430,688846502588400,4721424167835365,32361122672259150,221806434537978680,1520283919093591605
%N a(n) = Lucas(4n+3) + 1, or 5*Fibonacci(2n+1)*Fibonacci(2n+2).
%D Hugh C. Williams, Edouard Lucas and Primality Testing, John Wiley and Sons, 1998, p. 75.
%H G. C. Greubel, <a href="/A081015/b081015.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) = 8*a(n-1) - 8*a(n-2) + a(n-3).
%F G.f.: 5*(1-2*x)/((1-x)*(1-7*x+x^2)). - _Colin Barker_, Jun 24 2012
%p luc := proc(n) option remember: if n=0 then RETURN(2) fi: if n=1 then RETURN(1) fi: luc(n-1)+luc(n-2): end: for n from 0 to 25 do printf(`%d,`,luc(4*n+3)+1) od: # _James A. Sellers_, Mar 03 2003
%t LucasL[4*Range[0,20] +3] +1 (* _G. C. Greubel_, Jul 14 2019 *)
%t LinearRecurrence[{8,-8,1},{5,30,200},30] (* _Harvey P. Dale_, Dec 06 2021 *)
%o (PARI) vector(30, n, n--; f=fibonacci; f(4*n+4)+f(4*n+2)+1) \\ _G. C. Greubel_, Jul 14 2019
%o (Magma) [Lucas(4*n+3)+1: n in [0..30]]; // _G. C. Greubel_, Jul 14 2019
%o (Sage) [lucas_number2(4*n+3,1,-1)+1 for n in (0..30)] # _G. C. Greubel_, Jul 14 2019
%o (GAP) List([0..30], n-> Lucas(1,-1,4*n+3)[2] +1); # _G. C. Greubel_, Jul 14 2019
%o (Python)
%o from sympy import lucas
%o def a(n): return lucas(4*n+3) + 1
%o print([a(n) for n in range(22)]) # _Michael S. Branicky_, May 30 2021
%Y Cf. A000045 (Fibonacci numbers), A000032 (Lucas numbers).
%K nonn,easy
%O 0,1
%A _R. K. Guy_, Mar 01 2003
%E More terms from _James A. Sellers_, Mar 03 2003
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