

A121177


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


3



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
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OFFSET

1,2


COMMENTS

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 treelike octagonal systems: catapolyoctagons, ACH Models in Chem. 134 (1997), 5570, eq. (6).


LINKS



FORMULA

G.f.: 2*x^2 / ( (5*x1)*(x1) ).  R. J. Mathar, Jul 31 2019


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



