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%I #25 Jan 02 2024 08:52:14
%S 0,10,75,520,3570,24475,167760,1149850,7881195,54018520,370248450,
%T 2537720635,17393796000,119218851370,817138163595,5600748293800,
%U 38388099893010,263115950957275,1803423556807920,12360848946698170
%N a(n) = Lucas(4n+1) - 1, or 5*Fibonacci(2n)*Fibonacci(2n+1).
%D Hugh C. Williams, Edouard Lucas and Primality Testing, John Wiley and Sons, 1998, p. 75.
%H G. C. Greubel, <a href="/A081017/b081017.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*x*(2-x)/((1-x)*(1-7*x+x^2)). - _Colin Barker_, Apr 16 2012
%p with(combinat): 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+1)-1) od: # _James A. Sellers_, Mar 03 2003
%t LucasL[4*Range[0,20]+1]-1 (* or *) LinearRecurrence[{8,-8,1},{0,10,75},20] (* _Harvey P. Dale_, Mar 02 2015 *)
%o (PARI) vector(20, n, n--; f=fibonacci; f(4*n+2)+f(4*n)-1) \\ _G. C. Greubel_, Jul 14 2019
%o (Magma) [Lucas(4*n+1) -1: n in [0..20]]; // _G. C. Greubel_, Jul 14 2019
%o (Sage) [lucas_number2(4*n+1, 1,-1) - 1 for n in (0..20)] # _G. C. Greubel_, Jul 14 2019
%o (GAP) List([0..20], n-> Lucas(1,-1, 4*n+1)[2] -1 ); # _G. C. Greubel_, Jul 14 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 03 2003
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