A039658 Related to enumeration of edge-rooted catafusenes.

0, 1, 2, 5, 8, 18, 28, 64, 100, 237, 374, 917, 1460, 3679, 5898, 15183, 24468, 64055, 103642, 275011, 446380, 1197616, 1948852, 5277070, 8605288, 23483743, 38362198, 105392983, 172423768, 476459938, 780496108, 2167743688, 3554991268

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

From Petros Hadjicostas, Jan 13 2019: (Start)

This sequence appears in Table I, p. 533, in Cyvin et al. (1992) and Table I, p. 1174, in Cyvin et al. (1994).

In Cyvin et al. (1992), it is defined through eq. (22), p. 535. We have a(n) = Sum_{i=1..n-1} M(i)*M(n-i), where M(2*n) = M(2*n-1) = A007317(n) for n >= 1.

In Cyvin et al. (1992), it is used in the calculation of sequence A026118. See eq. (34), p. 536, in Cyvin et al. (1992).

(The word "annelated" in the title of Cyvin et al. (1994) is spelled "annealated" in the text of Cyvin et al. (1992).)

(End)

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• S. J. Cyvin, Zhang Fuji, B. N. Cyvin, Guo Xiaofeng, and J. Brunvoll, Enumeration and classification of benzenoid systems. 32. Normal perifusenes with two internal vertices, J. Chem. Inform. Comput. Sci., 32 (1992), 532-540.
• S. J. Cyvin, B. N. Cyvin, J. Brunvoll, and E. Brendsdal, Enumeration and classification of certain polygonal systems representing polycyclic conjugated hydrocarbons: annelated catafusenes, J. Chem. Inform. Comput. Sci., 34 (1994), 1174-1180.
• Eric Weisstein's World of Mathematics, Fusenes.
• Eric Weisstein's World of Mathematics, Polyhex.

Crossrefs

Cf. A007317, A026118.

Programs

Mathematica

Rest[CoefficientList[Series[(1+x) (1-3x^2-Sqrt[1-6x^2+5x^4])/(2x^2 (1-x)),{x,0,40}],x]] (* Harvey P. Dale, Oct 30 2016 *)

Formula

G.f.: (1+x)*(1 - 3*x^2 - sqrt(1 - 6*x^2 + 5*x^4))/(2*x^2*(1-x)) (eq. (9), p. 1175, in Cyvin et al. (1994)).

For n >= 1, a(n) = Sum_{i=1..n-1} A007317(floor((i+1)/2)) * A007317(floor((n-i+1)/2)). - Petros Hadjicostas, Jan 13 2019

Extensions

More terms from Emeric Deutsch, Mar 14 2004