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 A129743 #39 Jun 08 2025 01:13:14
%S A129743 2,12,50,192,722,2700,10082,37632,140450,524172,1956242,7300800,
%T A129743 27246962,101687052,379501250,1416317952,5285770562,19726764300,
%U A129743 73621286642,274758382272,1025412242450,3826890587532,14282150107682,53301709843200,198924689265122,742397047217292
%N A129743 a(n) = -(u^n-1)*(v^n-1) with u = 2+sqrt(3), v = 2-sqrt(3).
%C A129743 Each term of this sequence beyond the sixth has a primitive prime divisor. - Anthony Flatters (Anthony.Flatters(AT)uea.ac.uk), Aug 17 2007
%C A129743 a(n) is also the number of spanning trees for the n-gear graph. - _Eric W. Weisstein_, Jul 16 2011
%H A129743 Stefano Spezia, <a href="/A129743/b129743.txt">Table of n, a(n) for n = 1..1700</a>
%H A129743 G. Everest et al., <a href="http://www.jstor.org/stable/27642221">Primes generated by recurrence sequences</a>, Amer. Math. Monthly, 114 (No. 5, 2007), 417-431.
%H A129743 Anthony Flatters, <a href="https://arxiv.org/abs/0708.2190">Primitive Divisors of some Lehmer-Pierce Sequences</a>, arXiv:0708.2190 [math.NT], 2007.
%H A129743 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GearGraph.html">Gear Graph</a>
%H A129743 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SpanningTree.html">Spanning Tree</a>
%H A129743 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-5,1).
%F A129743 a(2*n) = 12*A001353(n)^2, a(2*n+1) = 2*A001834(n)^2. - _Vladeta Jovovic_, May 30 2007
%F A129743 a(n) = 2*A092184(n). - _Robert G. Wilson v_, Jul 04 2007
%F A129743 O.g.f.: 2*x*(1+x)/((1-x)*(1-4*x+x^2)). - _R. J. Mathar_, Dec 05 2007
%F A129743 a(n) = 5*a(n-1) - 5*a(n-2) + a(n-3). - _Eric W. Weisstein_, Jul 15 2011
%F A129743 E.g.f.: 2*exp(x)*(exp(x)*cosh(sqrt(3)*x) - 1). - _Stefano Spezia_, May 05 2024
%p A129743 u:=2+sqrt(3): v:=2-sqrt(3): a:=n->expand(-(u^n-1)*(v^n-1)): seq(a(n),n=1..28); # _Emeric Deutsch_, May 13 2007
%t A129743 Table[-((2 + Sqrt[3])^n - 1)*((2 - Sqrt[3])^n - 1), {n, 30}] // Expand (* _Stefan Steinerberger_, May 15 2007 *)
%t A129743 LinearRecurrence[{5, -5, 1}, {2, 12, 50}, 30]
%t A129743 LucasL[2 Range[20], Sqrt[2]] - 2 // Round (* _Eric W. Weisstein_, Mar 28 2018 *)
%o A129743 (PARI) my(x='x+O('x^30)); Vec(2*x*(1+x)/((1-x)*(1-4*x+x^2))) \\ _Altug Alkan_, Mar 28 2018
%Y A129743 Cf. A001353, A001834, A092184.
%K A129743 nonn,easy
%O A129743 1,1
%A A129743 _N. J. A. Sloane_, May 13 2007
%E A129743 More terms from _Emeric Deutsch_ and _Stefan Steinerberger_, May 13 2007
%E A129743 More terms from _Vladeta Jovovic_, May 30 2007