A000577 - OEIS

A000577

Number of triangular polyominoes (or triangular polyforms, or polyiamonds) with n cells (turning over is allowed, holes are allowed, must be connected along edges).
(Formerly M2374 N0941)

1, 1, 1, 3, 4, 12, 24, 66, 160, 448, 1186, 3334, 9235, 26166, 73983, 211297, 604107, 1736328, 5000593, 14448984, 41835738, 121419260, 353045291, 1028452717, 3000800627, 8769216722, 25661961898, 75195166667, 220605519559, 647943626796, 1905104762320, 5607039506627, 16517895669575

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

If holes are not allowed, we get A070765. - Joseph Myers, Apr 20 2009

It is a consequence of Madras's 1999 pattern theorem that almost all polyiamonds have holes, i.e., lim_{n->oo} A070765(n)/A000577(n) = 0. - Johann Peters, Jan 06 2024

REFERENCES

F. Harary, Graphical enumeration problems; in Graph Theory and Theoretical Physics, ed. F. Harary, Academic Press, London, 1967, pp. 1-41.

W. F. Lunnon, Counting hexagonal and triangular polyominoes, pp. 87-100 of R. C. Read, editor, Graph Theory and Computing. Academic Press, NY, 1972.

Ed Pegg, Jr., Polyform puzzles, in Tribute to a Mathemagician, Peters, 2005, pp. 119-125.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

P. J. Torbijn, Polyiamonds, J. Rec. Math., 2 (1969), 216-227.

LINKS

John Mason, Table of n, a(n) for n = 1..52

Ed Pegg, Jr., Illustrations of polyforms

A. Clarke, Polyiamonds

S. T. Coffin, The Puzzling World of Polyhedral Dissections, Chap 2, Table 1.

R. K. Guy, O'Beirne's Hexiamond, in The Mathemagician and the Pied Puzzler - A Collection in Tribute to Martin Gardner, Ed. E. R. Berlekamp and T. Rogers, A. K. Peters, 1999, 85-96 [broken link?]

J. and J. Hindriks, Dutchman Designs: Quilting Patterns [broken link?]

Kadon Enterprises, The 66 polyominoes of order 8 (from a puzzle)

Kadon Enterprises, Home page

M. Keller, Counting polyforms.

N. Madras, A pattern theorem for lattice clusters, arXiv:math/9902161 [math.PR], 1999; Annals of Combinatorics, 3 (1999), 357-384.

Greg Malen, Érika Roldán, and Rosemberg Toalá-Enríquez, Extremal {p, q}-Animals, Ann. Comb. (2023), p. 3.

John Mason, Counting Polyiamonds

R. J. Mathar, Illustrations for up to 9-iamonds

Jaime Rangel-Mondragon, Polyominoes and Related Families, The Mathematica Journal, 9:3 (2005), 609-640.

Eric Weisstein's World of Mathematics, Polyiamond

CROSSREFS

Cf. A000207, A006534, A030223, A030224, A001420, A096361, A070765.