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

1,2

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

Table of n, a(n) for n=1..25.

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).

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

Crossrefs

Cf. A121101, A125831.

Sequence in context: A037562 A321277 A187001 * A125831 A289787 A226506

Adjacent sequences: A121174 A121175 A121176 * A121178 A121179 A121180

Keyword

nonn,easy

Author

N. J. A. Sloane, Aug 15 2006

Status

approved