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%I M2374 N0941 #84 Jan 07 2024 01:38:47
%S 1,1,1,3,4,12,24,66,160,448,1186,3334,9235,26166,73983,211297,604107,
%T 1736328,5000593,14448984,41835738,121419260,353045291,1028452717,
%U 3000800627,8769216722,25661961898,75195166667,220605519559,647943626796,1905104762320,5607039506627,16517895669575
%N Number of triangular polyominoes (or triangular polyforms, or polyiamonds) with n cells (turning over is allowed, holes are allowed, must be connected along edges).
%C If holes are not allowed, we get A070765. - _Joseph Myers_, Apr 20 2009
%C 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
%D F. Harary, Graphical enumeration problems; in Graph Theory and Theoretical Physics, ed. F. Harary, Academic Press, London, 1967, pp. 1-41.
%D W. F. Lunnon, Counting hexagonal and triangular polyominoes, pp. 87-100 of R. C. Read, editor, Graph Theory and Computing. Academic Press, NY, 1972.
%D Ed Pegg, Jr., Polyform puzzles, in Tribute to a Mathemagician, Peters, 2005, pp. 119-125.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D P. J. Torbijn, Polyiamonds, J. Rec. Math., 2 (1969), 216-227.
%H John Mason, <a href="/A000577/b000577.txt">Table of n, a(n) for n = 1..52</a>
%H Ed Pegg, Jr., <a href="http://demonstrations.wolfram.com/PolyformExplorer/">Illustrations of polyforms</a>
%H A. Clarke, <a href="http://www.recmath.com/PolyPages/PolyPages/Polyiamonds.htm">Polyiamonds</a>
%H S. T. Coffin, <a href="http://www.johnrausch.com/PuzzlingWorld/chap02.htm#fig024">The Puzzling World of Polyhedral Dissections</a>, Chap 2, Table 1.
%H R. K. Guy, <a href="http://www.g4g4.com/contentsmmpp.html">O'Beirne's Hexiamond</a>, 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?]
%H J. and J. Hindriks, <a href="http://www.dutdes.com/ddfree.html">Dutchman Designs: Quilting Patterns</a> [broken link?]
%H Kadon Enterprises, <a href="/A000577/a000577.gif">The 66 polyominoes of order 8</a> (from a puzzle)
%H Kadon Enterprises, <a href="http://www.gamepuzzles.com/esspoly.htm#OR">Home page</a>
%H M. Keller, <a href="http://www.solitairelaboratory.com/polyenum.html">Counting polyforms</a>.
%H N. Madras, <a href="http://arxiv.org/abs/math/9902161">A pattern theorem for lattice clusters</a>, arXiv:math/9902161 [math.PR], 1999; Annals of Combinatorics, 3 (1999), 357-384.
%H Greg Malen, Érika Roldán, and Rosemberg Toalá-Enríquez, <a href="https://doi.org/10.1007/s00026-022-00631-1">Extremal {p, q}-Animals</a>, Ann. Comb. (2023), p. 3.
%H John Mason, <a href="/A000577/a000577_1.pdf">Counting Polyiamonds</a>
%H R. J. Mathar, <a href="/A000577/a000577.pdf">Illustrations for up to 9-iamonds</a>
%H Jaime Rangel-Mondragon, <a href="https://web.archive.org/web/20190411024906/http://www.mathematica-journal.com/issue/v9i3/polyominoes.html">Polyominoes and Related Families</a>, The Mathematica Journal, 9:3 (2005), 609-640.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Polyiamond.html">Polyiamond</a>
%Y Cf. A000207, A006534, A030223, A030224, A001420, A096361, A070765.
%Y Cf. also A000105, A000228, A103465.
%K nonn,hard,nice
%O 1,4
%A _N. J. A. Sloane_
%E More terms from _David W. Wilson_
%E a(19) from _Achim Flammenkamp_, Feb 15 1999
%E a(20), a(21), a(22), a(23) and a(24) from Brendan Owen (brendan_owen(AT)yahoo.com), Jan 01 2002
%E a(25) to a(28) from _Joseph Myers_, Sep 24 2002
%E Link updated by _William Rex Marshall_, Dec 16 2009
%E a(29) and a(30) from _Joseph Myers_, Nov 21 2010
%E More terms from _John Mason_, Oct 28 2023
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