This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A081003 #35 Jul 02 2025 16:02:01
%S A081003 2,6,35,234,1598,10947,75026,514230,3524579,24157818,165580142,
%T A081003 1134903171,7778742050,53316291174,365435296163,2504730781962,
%U A081003 17167680177566,117669030460995,806515533049394,5527939700884758,37889062373143907,259695496911122586
%N A081003 a(n) = Fibonacci(4n+1) + 1, or Fibonacci(2n+1)*Lucas(2n).
%D A081003 Hugh C. Williams, Edouard Lucas and Primality Testing, John Wiley and Sons, 1998, p. 75
%H A081003 Nathaniel Johnston, <a href="/A081003/b081003.txt">Table of n, a(n) for n = 0..400</a>
%H A081003 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (8,-8,1).
%F A081003 a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3).
%F A081003 G.f.: (2-10*x+3*x^2)/((1-x)*(1-7*x+x^2)). - _Colin Barker_, Jun 24 2012
%F A081003 Product_{n>=0} (1 + 1/a(n)) = (2 + phi)/2 (A296182). - _Amiram Eldar_, Nov 28 2024
%p A081003 with(combinat): for n from 0 to 30 do printf(`%d,`,fibonacci(4*n+1)+1) od: # _James Sellers_, Mar 03 2003
%t A081003 Fibonacci[4*Range[0,30]+1]+1 (* or *) LinearRecurrence[{8,-8,1}, {2,6,35}, 30] (* _Harvey P. Dale_, Jul 20 2011 *)
%o A081003 (Magma) [Fibonacci(4*n+1) +1: n in [0..30]]; // _Vincenzo Librandi_, Apr 15 2011
%o A081003 (PARI) vector(30, n, n--; fibonacci(4*n+1)+1) \\ _G. C. Greubel_, Jul 15 2019
%o A081003 (Sage) [fibonacci(4*n+1)+1 for n in (0..30)] # _G. C. Greubel_, Jul 15 2019
%o A081003 (GAP) List([0..30], n-> Fibonacci(4*n+1)+1); # _G. C. Greubel_, Jul 15 2019
%Y A081003 Cf. A000045 (Fibonacci numbers), A000032 (Lucas numbers), A296182.
%K A081003 nonn,easy
%O A081003 0,1
%A A081003 _R. K. Guy_, Mar 01 2003
%E A081003 More terms from _James Sellers_, Mar 03 2003