|
%I #68 Dec 24 2023 09:37:30
%S 0,4,40,396,3920,38804,384120,3802396,37639840,372596004,3688320200,
%T 36510605996,361417739760,3577666791604,35415250176280,
%U 350574834971196,3470333099535680,34352756160385604,340057228504320360,3366219528882817996,33322138060323859600
%N a(0) = 0, a(1) = 4; for n > 1, a(n) = 10*a(n-1) - a(n-2).
%C Kekulé numbers for the benzenoids P_2(n).
%C 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
%C Numbers n such that 6*n^2 + 4 is a square. - _Colin Barker_, Mar 17 2014
%D 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)}).
%H Michael De Vlieger, <a href="/A122652/b122652.txt">Table of n, a(n) for n = 0..1004</a>
%H Andersen, K., Carbone, L. and Penta, D., <a href="https://pdfs.semanticscholar.org/8f0c/c3e68d388185129a56ed73b5d21224659300.pdf">Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields</a>, Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245-278, 2011. See Section 9.
%H Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Nemeth/nemeth7.html">Ellipse Chains and Associated Sequences</a>, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
%H John M. Campbell, <a href="http://arxiv.org/abs/1105.3399">An Integral Representation of Kekulé Numbers, and Double Integrals Related to Smarandache Sequences</a>, arXiv preprint arXiv:1105.3399 [math.GM], 2011.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10,-1).
%F G.f.: 4*x/(1 - 10*x + x^2). - _Philippe Deléham_, Nov 17 2008
%F 3*a(n)^2 + 2 = 2*A001079(n)^2. - _Charlie Marion_, Feb 01 2013
%F a(n) = (2*arcsinh(sqrt(2))*sinh(2*n*arcsinh(sqrt(2)))/log(sqrt(2) + sqrt(3)))/sqrt(6). - _Artur Jasinski_, Aug 09 2016
%F a(n) = 2*A001078(n). - _Bruno Berselli_, Nov 25 2016
%F E.g.f.: sqrt(6)*exp(5*x)*sinh(2*sqrt(6)*x)/3. - _Franck Maminirina Ramaharo_, Jan 07 2019
%t CoefficientList[Series[(4 z)/(z^2 - 10 z + 1), {z, 0, 200}], z] (* _Vladimir Joseph Stephan Orlovsky_, Jun 11 2011 *)
%t LinearRecurrence[{10, -1}, {0, 4}, 21] (* _Jean-François Alcover_, Jan 07 2019 *)
%o (PARI) a(n)=if(n<2,(n%2)*4,10*a(n-1)-a(n-2)) \\ _Benoit Cloitre_, Sep 23 2006
%Y Cf. A001078, A001079, A032528.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Sep 21 2006
%E More terms and better definition from _Benoit Cloitre_, Sep 23 2006
|
