%I #35 May 05 2024 20:02:22

%S 2,12,50,192,722,2700,10082,37632,140450,524172,1956242,7300800,

%T 27246962,101687052,379501250,1416317952,5285770562,19726764300,

%U 73621286642,274758382272,1025412242450,3826890587532,14282150107682,53301709843200,198924689265122,742397047217292

%N a(n) = -(u^n-1)*(v^n-1) with u = 2+sqrt(3), v = 2-sqrt(3).

%C 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 a(n) is also the number of spanning trees for the n-gear graph. - _Eric W. Weisstein_, Jul 16 2011

%H Stefano Spezia, <a href="/A129743/b129743.txt">Table of n, a(n) for n = 1..1700</a>

%H 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 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GearGraph.html">Gear Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SpanningTree.html">Spanning Tree</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-5,1).

%F a(2*n) = 12*A001353(n)^2, a(2*n+1) = 2*A001834(n)^2. - _Vladeta Jovovic_, May 30 2007

%F a(n) = 2*A092184(n). - _Robert G. Wilson v_, Jul 04 2007

%F O.g.f.: 2*x*(1+x)/((1-x)*(1-4*x+x^2)). - _R. J. Mathar_, Dec 05 2007

%F a(n) = 5*a(n-1) - 5*a(n-2) + a(n-3). - _Eric W. Weisstein_, Jul 15 2011

%F E.g.f.: 2*exp(x)*(exp(x)*cosh(sqrt(3)*x) - 1). - _Stefano Spezia_, May 05 2024

%p 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 Table[-((2 + Sqrt[3])^n - 1)*((2 - Sqrt[3])^n - 1)], {n, 30}] // Expand (* _Stefan Steinerberger_, May 15 2007 *)

%t LinearRecurrence[{5, -5, 1}, {2, 12, 50}, 30]

%t LucasL[2 Range[20], Sqrt[2]] - 2 // Round (* _Eric W. Weisstein_, Mar 28 2018 *)

%o (PARI) x='x+O('x^99); Vec(2*x*(1+x)/((1-x)*(1-4*x+x^2))) \\ _Altug Alkan_, Mar 28 2018

%Y Cf. A001353, A001834, A092184.

%K nonn,easy

%O 1,1

%A _N. J. A. Sloane_, May 13 2007

%E More terms from _Emeric Deutsch_ and _Stefan Steinerberger_, May 13 2007

%E More terms from _Vladeta Jovovic_, May 30 2007