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A081069 a(n) = Lucas(4n)+2 = Lucas(2n)^2.

%I A081069 #58 Jul 02 2025 16:02:01
%S A081069 4,9,49,324,2209,15129,103684,710649,4870849,33385284,228826129,
%T A081069 1568397609,10749957124,73681302249,505019158609,3461452808004,
%U A081069 23725150497409,162614600673849,1114577054219524,7639424778862809
%N A081069 a(n) = Lucas(4n)+2 = Lucas(2n)^2.
%D A081069 Hugh C. Williams, Edouard Lucas and Primality Testing, John Wiley and Sons, 1998, p. 75.
%H A081069 Pridon Davlianidze, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Problems/ElemProbMay2020.pdf">Problem B-1270</a>, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 58, No. 2 (2020), p. 179; <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Problems/FQElemProbMay2021.pdf">Four Telescopic Infinite Products</a>, Solution to Problem B-1270 by Jason L. Smith, ibid., Vol. 59, No. 2 (2021), pp. 183-184.
%H A081069 Emrah Kılıç, Yücel Türker Ulutaş, and Neşe Ömür, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Omur/omur6.html">A Formula for the Generating Functions of Powers of Horadam's Sequence with Two Additional Parameters</a>, J. Int. Seq. 14 (2011) #11.5.6, Table 2, k=2.
%H A081069 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (8,-8,1).
%F A081069 a(n) = A005248(n)^2 = A056854(n)+2.
%F A081069 a(n) = 8a(n-1) - 8a(n-2) + a(n-3).
%F A081069 a(n) = 2^(4*n)*(cos(Pi/5)^(2*n)+cos(3*Pi/5)^(2*n))^2. - _Gary Detlefs_, Dec 05 2010
%F A081069 From _Gary Detlefs_, Dec 06 2010: (Start)
%F A081069 a(n) = 7*a(n-1)-a(n-2)-10, n>1.
%F A081069 a(n) = 5*Sum_{k=0..n}(Fibonacci(4*k+2))+4, with offset -1. (End)
%F A081069 G.f.: -(9*x^2-23*x+4)/((x-1)*(x^2-7*x+1)). - _Colin Barker_, Jun 24 2012
%F A081069 Product_{n>=0} (1 + 5/a(n)) = 3*phi^2/2, where phi is the golden ratio (A001622) (Davlianidze, 2020). - _Amiram Eldar_, Dec 04 2024
%F A081069 a(n) = Sum_{k>=0} Lucas(2*n*k)/(Lucas(2*n)^k). - _Diego Rattaggi_, Jan 12 2025
%F A081069 E.g.f.: 2*(cosh(x) + exp(7*x/2)*cosh(3*sqrt(5)*x/2) + sinh(x)). - _Stefano Spezia_, Jan 20 2025
%p A081069 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 40 do printf(`%d,`,luc(4*n)+2) od: # _James Sellers_, Mar 05 2003
%p A081069 G:=(x,n)-> cos(x)^n +cos(3*x)^n: seq(simplify(2^(4*n)*G(Pi/5,2*n)^2), n=0..19) # _Gary Detlefs_, Dec 05 2010
%p A081069 t:= n-> sum(fibonacci(4*k+2),k=0..n):seq(5*t(n)+4,n=-1..18); # _Gary Detlefs_, Dec 06 2010
%t A081069 LucasL[4*Range[0,20]]+2 (* _Harvey P. Dale_, Sep 09 2012 *)
%o A081069 (Magma) [ Lucas(2*n)^2: n in [0..70] ]; // _Vincenzo Librandi_, Apr 16 2011
%Y A081069 Cf. A000032 (Lucas numbers), A001622, A005248, A056854.
%K A081069 nonn,easy
%O A081069 0,1
%A A081069 _R. K. Guy_, Mar 04 2003