A003204 Cluster series for honeycomb. (Formerly M2557)

1, 3, 6, 12, 24, 33, 60, 99, 156, 276, 438, 597, 1134, 1404, 2904, 3522, 6876, 7548, 16680, 18153, 39846, 41805

Offset

0,2

Comments

The word "cluster" here essentially means polyiamond. This sequence can be computed based on a calculation of the perimeter polynomials of polyiamonds. In particular, if P_n(x) is the perimeter polynomial for all fixed polyiamonds of size n, then this sequence is the coefficients of x in Sum_{k>=1} k^2 * x^k * P_k(1-x). - Sean A. Irvine, Aug 16 2020

References

J. W. Essam, Percolation and cluster size, in C. Domb and M. S. Green, Phase Transitions and Critical Phenomena, Ac. Press 1972, Vol. 2; see especially pp. 225-226.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Table of n, a(n) for n=0..21.

Sean A. Irvine, Java program (github)

M. F. Sykes and J. W. Essam, Critical percolation probabilities by series methods, Phys. Rev., 133 (1964), A310-A315.

M. F. Sykes and M. Glen, Percolation processes in two dimensions. I. Low-density series expansions, J. Phys. A: Math. Gen., 9 (1976), 87-95.

Crossrefs

Cf. A001420, A003202 (triangular net), A003203 (square net), A003199 (bond percolation).

Sequence in context: A319445 A316318 A173216 * A038620 A250300 A363692

Adjacent sequences: A003201 A003202 A003203 * A003205 A003206 A003207

Keyword

nonn,more

Author

N. J. A. Sloane

Extensions

a(12)-a(18) from Sean A. Irvine, Aug 16 2020

a(19)-a(21) added from Sykes & Glen by Andrey Zabolotskiy, Feb 01 2022

Status

approved