A129743 a(n) = -(u^n-1)*(v^n-1) with u = 2+sqrt(3), v = 2-sqrt(3).

2, 12, 50, 192, 722, 2700, 10082, 37632, 140450, 524172, 1956242, 7300800, 27246962, 101687052, 379501250, 1416317952, 5285770562, 19726764300, 73621286642, 274758382272, 1025412242450, 3826890587532, 14282150107682, 53301709843200, 198924689265122, 742397047217292

Offset: 1

Author

N. J. A. Sloane, May 13 2007

Keywords

• nonn
• easy

Comments

Each term of this sequence beyond the sixth has a primitive prime divisor. - Anthony Flatters (Anthony.Flatters(AT)uea.ac.uk), Aug 17 2007

a(n) is also the number of spanning trees for the n-gear graph. - Eric W. Weisstein, Jul 16 2011

Links

• Stefano Spezia, Table of n, a(n) for n = 1..1700
• G. Everest et al., Primes generated by recurrence sequences, Amer. Math. Monthly, 114 (No. 5, 2007), 417-431.
• Anthony Flatters, Primitive Divisors of some Lehmer-Pierce Sequences, arXiv:0708.2190 [math.NT], 2007.
• Eric Weisstein's World of Mathematics, Gear Graph
• Eric Weisstein's World of Mathematics, Spanning Tree
• Index entries for linear recurrences with constant coefficients, signature (5,-5,1).

Crossrefs

Cf. A001353, A001834, A092184.

Programs

Maple

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

Mathematica

Table[-((2 + Sqrt[3])^n - 1)*((2 - Sqrt[3])^n - 1), {n, 30}] // Expand (* Stefan Steinerberger, May 15 2007 *)
LinearRecurrence[{5, -5, 1}, {2, 12, 50}, 30]
LucasL[2 Range[20], Sqrt[2]] - 2 // Round (* Eric W. Weisstein, Mar 28 2018 *)

PARI

my(x='x+O('x^30)); Vec(2*x*(1+x)/((1-x)*(1-4*x+x^2))) \\ Altug Alkan, Mar 28 2018

Formula

a(2*n) = 12*A001353(n)^2, a(2*n+1) = 2*A001834(n)^2. - Vladeta Jovovic, May 30 2007

a(n) = 2*A092184(n). - Robert G. Wilson v, Jul 04 2007

O.g.f.: 2*x*(1+x)/((1-x)*(1-4*x+x^2)). - R. J. Mathar, Dec 05 2007

a(n) = 5*a(n-1) - 5*a(n-2) + a(n-3). - Eric W. Weisstein, Jul 15 2011

E.g.f.: 2*exp(x)*(exp(x)*cosh(sqrt(3)*x) - 1). - Stefano Spezia, May 05 2024

Extensions

More terms from Emeric Deutsch and Stefan Steinerberger, May 13 2007

More terms from Vladeta Jovovic, May 30 2007