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%I #33 Jan 02 2024 08:52:04
%S 3,7,36,235,1599,10948,75027,514231,3524580,24157819,165580143,
%T 1134903172,7778742051,53316291175,365435296164,2504730781963,
%U 17167680177567,117669030460996,806515533049395,5527939700884759,37889062373143908,259695496911122587
%N a(n) = Fibonacci(4n+1) + 2, or Fibonacci(2n-1)*Lucas(2n+2).
%D Hugh C. Williams, Edouard Lucas and Primality Testing, John Wiley and Sons, 1998, p. 75.
%H Nathaniel Johnston, <a href="/A081010/b081010.txt">Table of n, a(n) for n = 0..500</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 a(n) = 2 + (A001906(n+1)^2 + A001519(n)^2)/2. - _Creighton Dement_, Aug 15 2004
%F G.f.: (3-17*x+4*x^2)/((1-x)*(1-7*x+x^2)). - _Colin Barker_, Jun 24 2012
%p with(combinat) for n from 0 to 30 do printf(`%d,`,fibonacci(4*n+1)+2) od # _James A. Sellers_, Mar 03 2003
%t Fibonacci[4*Range[0,30]+1]+2 (* _G. C. Greubel_, Jul 14 2019 *)
%o (Magma) [Fibonacci(4*n+1) +2: n in [0..30]]; // _Vincenzo Librandi_, Apr 15 2011
%o (PARI) vector(30, n, n--; fibonacci(4*n+1)+2) \\ _G. C. Greubel_, Jul 14 2019
%o (Sage) [fibonacci(4*n+1)+2 for n in (0..30)] # _G. C. Greubel_, Jul 14 2019
%o (GAP) List([0..30], n-> Fibonacci(4*n+1)+2); # _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
%E More terms from _James A. Sellers_, Mar 03 2003
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