

A121177


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


2



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

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 space 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

Table of n, a(n) for n=1..25.
P. E. Weidmann, The OEIS Sequencer survey, Apr 11 2015
Wikipedia, Space group.
Index entries for linear recurrences with constant coefficients, signature (6,5).


FORMULA

a(n) = (5^n5)/10 = 2*A003463(n1) for n >= 1.  Philipp Emanuel Weidmann, cf. link.
G.f.: 2*x^2 / ( (5*x1)*(x1) ).  R. J. Mathar, Jul 31 2019


CROSSREFS

Cf. A121101, A125831.
Sequence in context: A321277 A187001 A125831 * A289787 A226506 A026076
Adjacent sequences: A121174 A121175 A121176 * A121178 A121179 A121180


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane, Aug 15 2006


STATUS

approved



