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a(n) = Lucas(4*n+1) + 1, or Lucas(2*n)*Lucas(2*n+1).
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%I #42 Nov 08 2024 15:12:26

%S 2,12,77,522,3572,24477,167762,1149852,7881197,54018522,370248452,

%T 2537720637,17393796002,119218851372,817138163597,5600748293802,

%U 38388099893012,263115950957277,1803423556807922,12360848946698172

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

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

%H G. C. Greubel, <a href="/A081014/b081014.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 From _R. J. Mathar_, Sep 03 2010: (Start)

%F G.f.: (2 -4*x -3*x^2)/((1-x)*(1-7*x+x^2)).

%F a(n) = 1 + A056914(n). (End)

%F a(n) = 7*a(n-1) - a(n-2) - 5, n >= 2. - _R. J. Mathar_, Nov 07 2015

%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+1)+1) od: # _James A. Sellers_, Mar 03 2003

%t LinearRecurrence[{8,-8,1}, {2,12,77}, 20] (* _G. C. Greubel_, Dec 24 2017 *)

%t LucasL[4*Range[0,20] +1] +1 (* _G. C. Greubel_, Jul 14 2019 *)

%t CoefficientList[Series[(2-4x-3x^2)/((1-x)(1-7x+x^2)),{x,0,30}],x] (* _Harvey P. Dale_, Aug 27 2021 *)

%o (PARI) vector(20, n, n--; f=fibonacci; f(4*n+2)+f(4*n)+1) \\ _G. C. Greubel_, Dec 24 2017

%o (Magma) I:=[2,12,77]; [n le 3 select I[n] else 8*Self(n-1) - 8*Self(n-2) + Self(n-3): n in [0..30]]; // _G. C. Greubel_, Dec 24 2017

%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 0,1

%A _R. K. Guy_, Mar 01 2003