A121177 Catapolyoctagons (see Cyvin et al. for precise definition).

0, 2, 12, 62, 312, 1562, 7812, 39062, 195312, 976562, 4882812, 24414062, 122070312, 610351562, 3051757812, 15258789062, 76293945312, 381469726562, 1907348632812, 9536743164062, 47683715820312, 238418579101562, 1192092895507812, 5960464477539062, 29802322387695312

Offset: 1

Author

N. J. A. Sloane, Aug 15 2006

Keywords

• nonn
• easy

Comments

From Petros Hadjicostas, Jul 30 2019: (Start)

The conjecture by Philipp Emanuel Weidmann (see link below) is correct. In Cyvin et al. (1997), this sequence has a double meaning. See Eqs. (6) and (7) and Table I on p. 58 in that paper. The terms of the sequence are related to the enumeration of unbranched catapolyoctagons.

The number of unbranched catapolyoctagons of the symmetry C_{2h} is given by c_r = (1/2) *(5^(floor(r/2)-1) - 1) + (2/5) * binomial(1, r), where r is the number of octagons in the unbranched catapolyoctagon. We get the sequence 0, 0, 0, 2, 2, 12, 12, 62, 62, 312, 312, ... whose bijection (apart for the case r = 1) is the current sequence.

In addition, the number of unbranched catapolyoctagons of the symmetry C_{2v} is given by m_r = (1/2) * (3 - 2*(-1)^r) * 5^(floor(r/2) - 1) - (1/2), where again r is the number of octagons. We get the sequence 0, 0, 2, 2, 12, 12, 62, 62, 312, 312, 1562, 1562, ... whose bijection is the current sequence.

The total number of unbranched catapolygons (with respect to all the symmetry point groups D_{8h}, D_{2h}, C_{2h}, and C_{2v}) is given by i_r = A121101(r).

(End)

References

• S. J. Cyvin, B. N. Cyvin, and J. Brunvoll. Enumeration of tree-like octagonal systems: catapolyoctagons, ACH Models in Chem. 134 (1997), 55-70, eq. (6).

Links

• P. E. Weidmann, The OEIS Sequencer survey, Apr 11 2015.
• Wikipedia, Point groups in three dimensions.
• Index entries for linear recurrences with constant coefficients, signature (6,-5).

Crossrefs

Cf. A121101, A125831.

Formula

a(n) = (5^n-5)/10 = 2*A003463(n-1) for n >= 1. - Philipp Emanuel Weidmann, cf. link.

G.f.: 2*x^2 / ( (5*x-1)*(x-1) ). - R. J. Mathar, Jul 31 2019