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

1,1

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).

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

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 *)

Prog

(PARI) x='x+O('x^99); Vec(2*x*(1+x)/((1-x)*(1-4*x+x^2))) \\ Altug Alkan, Mar 28 2018

Crossrefs

Cf. A001353, A001834, A092184.

Sequence in context: A357782 A259802 A202789 * A115243 A218776 A241683

Adjacent sequences: A129740 A129741 A129742 * A129744 A129745 A129746

Keyword

nonn,easy

Author

N. J. A. Sloane, May 13 2007

Extensions

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

More terms from Vladeta Jovovic, May 30 2007

Status

approved