A003204 Cluster series for honeycomb.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• more

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

• 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).

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