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A122652
a(0) = 0, a(1) = 4; for n > 1, a(n) = 10*a(n-1) - a(n-2).
4
0, 4, 40, 396, 3920, 38804, 384120, 3802396, 37639840, 372596004, 3688320200, 36510605996, 361417739760, 3577666791604, 35415250176280, 350574834971196, 3470333099535680, 34352756160385604, 340057228504320360, 3366219528882817996, 33322138060323859600
OFFSET
0,2
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
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
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
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 *)
PROG
(PARI) a(n)=if(n<2, (n%2)*4, 10*a(n-1)-a(n-2)) \\ Benoit Cloitre, Sep 23 2006
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Sep 21 2006
EXTENSIONS
More terms and better definition from Benoit Cloitre, Sep 23 2006
STATUS
approved