A122653 a(n) = 10*a(n-1) - a(n-2) with a(0)=0, a(1)=6.

0, 6, 60, 594, 5880, 58206, 576180, 5703594, 56459760, 558894006, 5532480300, 54765908994, 542126609640, 5366500187406, 53122875264420, 525862252456794, 5205499649303520, 51529134240578406, 510085842756480540, 5049329293324226994, 49983207090485789400

Offset: 0

Author

N. J. A. Sloane, Sep 21 2006

Keywords

• nonn
• easy

Comments

Kekulé numbers for the benzenoids P''(n).

a(n) are the integer square roots of A032528(m) - 1. A001079 gives the value of m where these roots occur. Also see A122652. - Richard R. Forberg, Aug 05 2013

Numbers n such that 6*n^2 + 9 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. 301).

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (10,-1).

Programs

Mathematica

CoefficientList[Series[(6 z)/(z^2 - 10 z + 1), {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 11 2011 *)
LinearRecurrence[{10,-1},{0,6},30] (* Harvey P. Dale, Dec 16 2014 *)

PARI

a(n)=if(n<2,(n%2)*6,10*a(n-1)-a(n-2)) \\ Benoit Cloitre, Sep 23 2006

Formula

G.f.: 6x/(1 - 10x + x^2). - Philippe Deléham, Nov 17 2008

a(n) = 6*A004189(n). - R. J. Mathar, Jun 22 2020

6*a(n)^2+9 = (3*A001079(n))^2 - detail of the Barker comment. - R. J. Mathar, Jun 22 2020

Extensions

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