A122652 a(0) = 0, a(1) = 4; for n > 1, a(n) = 10*a(n-1) - a(n-2).

0, 4, 40, 396, 3920, 38804, 384120, 3802396, 37639840, 372596004, 3688320200, 36510605996, 361417739760, 3577666791604, 35415250176280, 350574834971196, 3470333099535680, 34352756160385604, 340057228504320360, 3366219528882817996, 33322138060323859600

Offset: 0

Author

N. J. A. Sloane, Sep 21 2006

Keywords

• nonn
• easy

Comments

Kekulé numbers for the benzenoids P_2(n).

a(n) are the values of m where A032528(m) - 1 has integer square roots. The roots are given by A001079. - Richard R. Forberg, Aug 05 2013

Numbers n such that 6*n^2 + 4 is a square. - Colin Barker, Mar 17 2014

References

• S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 283, K{P_2(n)}).

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..1004
• Andersen, K., Carbone, L. and Penta, D., Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields, Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245-278, 2011. See Section 9.
• Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
• John M. Campbell, An Integral Representation of Kekulé Numbers, and Double Integrals Related to Smarandache Sequences, arXiv preprint arXiv:1105.3399 [math.GM], 2011.
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (10,-1).

Crossrefs

Cf. A001078, A001079, A032528.

Programs

Mathematica

CoefficientList[Series[(4 z)/(z^2 - 10 z + 1), {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 11 2011 *)
LinearRecurrence[{10, -1}, {0, 4}, 21] (* Jean-François Alcover, Jan 07 2019 *)

PARI

a(n)=if(n<2,(n%2)*4,10*a(n-1)-a(n-2)) \\ Benoit Cloitre, Sep 23 2006

Formula

G.f.: 4*x/(1 - 10*x + x^2). - Philippe Deléham, Nov 17 2008

3*a(n)^2 + 2 = 2*A001079(n)^2. - Charlie Marion, Feb 01 2013

a(n) = (2*arcsinh(sqrt(2))*sinh(2*n*arcsinh(sqrt(2)))/log(sqrt(2) + sqrt(3)))/sqrt(6). - Artur Jasinski, Aug 09 2016

a(n) = 2*A001078(n). - Bruno Berselli, Nov 25 2016

E.g.f.: sqrt(6)*exp(5*x)*sinh(2*sqrt(6)*x)/3. - Franck Maminirina Ramaharo, Jan 07 2019

Extensions

More terms and better definition from Benoit Cloitre, Sep 23 2006