A000577 -id:A000577 - cp's OEIS frontend

A000105 Number of free polyominoes (or square animals) with n cells.

1, 1, 1, 2, 5, 12, 35, 108, 369, 1285, 4655, 17073, 63600, 238591, 901971, 3426576, 13079255, 50107909, 192622052, 742624232, 2870671950, 11123060678, 43191857688, 168047007728, 654999700403, 2557227044764, 9999088822075, 39153010938487, 153511100594603

Offset: 0

N. J. A. Sloane

For n>0, a(n) + A030228(n) = A000988(n) because the number of free polyominoes plus the number of polyominoes lacking bilateral symmetry equals the number of one-sided polyominoes. - Graeme McRae, Jan 05 2006
The possible symmetry groups of a (nonempty) polyomino are the 10 subgroups of the dihedral group D_8 of order 8: D_8, 1, Z_2 (five times), Z_4, (Z_2)^2 (twice). - Benoit Jubin, Dec 30 2008
Names for first few polyominoes: monomino, domino, tromino, tetromino, pentomino, hexomino, heptomino, octomino, enneomino, decomino, hendecomino, dodecomino, ...
Limit_{n->oo} a(n)^(1/n) = mu with 3.98 < mu < 4.64 (quoted by Castiglione et al., with a reference to Barequet et al., 2006, for the lower bound). The upper bound is due to Klarner and Rivest, 1973. By Madras, 1999, it is now known that this limit, also known as Klarner's constant, is equal to the limit growth rate lim_{n->oo} a(n+1)/a(n).
Polyominoes are worth exploring in the elementary school classroom. Students in grade 2 can reproduce the first 6 terms. Grade 3 students can explore area and perimeter. Grade 4 students can talk about polyomino symmetries.
The pentominoes should be singled out for special attention: 1) they offer a nice, manageable set that a teacher can commercially acquire without too much expense. 2) There are also deeply strategic games and perplexing puzzles that are great for all students. 3) A fraction of students will become engaged because of the beautiful solutions.
Conjecture: Almost all polyominoes are holey. In other words, A000104(n)/a(n) tends to 0 for increasing n. - John Mason, Dec 11 2021 (This is true, a consequence of Madras's 1999 pattern theorem. - Johann Peters, Jan 06 2024)

			a(0) = 1 as there is 1 empty polyomino with #cells = 0. - _Fred Lunnon_, Jun 24 2020

S. W. Golomb, Polyominoes, Appendix D, p. 152; Princeton Univ. Pr. NJ 1994
J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, CRC Press, 1997, p. 229.
D. A. Klarner, The Mathematical Gardner, p. 252 Wadsworth Int. CA 1981
W. F. Lunnon, Counting polyominoes, pp. 347-372 of A. O. L. Atkin and B. J. Birch, editors, Computers in Number Theory. Academic Press, NY, 1971.
W. F. Lunnon, Counting hexagonal and triangular polyominoes, pp. 87-100 of R. C. Read, editor, Graph Theory and Computing. Academic Press, NY, 1972.
George E. Martin, Polyominoes - A Guide to Puzzles and Problems in Tiling, The Mathematical Association of America, 1996
Ed Pegg, Jr., Polyform puzzles, in Tribute to a Mathemagician, Peters, 2005, pp. 119-125.
R. C. Read, Some applications of computers in graph theory, in L. W. Beineke and R. J. Wilson, editors, Selected Topics in Graph Theory, Academic Press, NY, 1978, pp. 417-444.
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).

John Mason, Table of n, a(n) for n = 0..50 (terms 0..45,47 from Toshihiro Shirakawa)
Z. Abel, E. Demaine, M. Demaine, H. Matsui and G. Rote, Common Developments of Several Different Orthogonal Boxes.
G. Barequet, M. Moffie, A. Ribo and G. Rote, Counting polyominoes on twisted cylinders, Integers 6 (2006), A22, 37 pp. (electronic).
K. S. Brown, Polyomino Enumerations
G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, Combinatorial aspects of L-convex polyominoes, European J. Combin. 28 (2007), no. 6, 1724-1741.
Juris Čerņenoks and Andrejs Cibulis, Tetrads and their Counting, Baltic J. Modern Computing, Vol. 6 (2018), No. 2, 96-106.
A. Clarke, Polyominoes
A. R. Conway and A. J. Guttmann, On two-dimensional percolation, J. Phys. A: Math. Gen. 28(1995) 891-904.
I. Jensen, Enumerations of lattice animals and trees, arXiv:cond-mat/0007239 [cond-mat.stat-mech], 2000.
I. Jensen and A. J. Guttmann, Statistics of lattice animals (polyominoes) and polygons, Journal of Physics A: Mathematical and General, vol. 33, pp. L257-L263, 2000.
M. Keller, Counting polyforms.
D. A. Klarner and R. L. Rivest, A procedure for improving the upper bound for the number of n-ominoes, Canadian J. of Mathematics, 25 (1973), 585-602.
N. Madras, A pattern theorem for lattice clusters, arXiv:math/9902161 [math.PR], 1999; Annals of Combinatorics, 3 (1999), 357-384.
John Mason, Counting size 50 polyominoes -V2
S. Mertens, Lattice animals: a fast enumeration algorithm and new perimeter polynomials, J. Statistical Physics, vol. 58, no. 5/6, pp. 1095-1108, Mar. 1990.
Stephan Mertens and Markus E. Lautenbacher, Counting lattice animals: A parallel attack J. Stat. Phys., vol. 66, no. 1/2, pp. 669-678, 1992.
W. R. Muller, K. Szymanski, J. V. Knop, and N. Trinajstic, On the number of square-cell configurations, Theor. Chim. Acta 86 (1993) 269-278
Joseph Myers, Polyomino tiling
Tomás Oliveira e Silva, Animal enumerations on regular tilings in Spherical, Euclidean and Hyperbolic 2-dimensional spaces
Tomás Oliveira e Silva, Animal enumerations on the {4,4} Euclidean tiling [The enumeration to order 28]
T. R. Parkin, L. J. Lander, and D. R. Parkin, Polyomino Enumeration Results, presented at SIAM Fall Meeting, 1967, and accompanying letter from T. J. Lander (annotated scanned copy)
Anuj Pathania, Scalable Task Schedulers for Many-Core Architectures, Ph.D. Thesis, Karlsruher Instituts für Technologie (Germany, 2018).
Ed Pegg, Jr., Illustrations of polyforms
Henri Picciotto, Polyomino Lessons
Jaime Rangel-Mondragón, Polyominoes and Related Families, The Mathematica Journal, Volume 9, Issue 3.
D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.
D. H. Redelmeier, Table 3 of Counting polyominoes...
Toshihiro Shirakawa, Harmonic Magic Square, pp 3-4: Enumeration of Polyominoes considering the symmetry, April 2012.
Herman Tulleken, Polyominoes 2.2: How they fit together, (2019).
Eric Weisstein's World of Mathematics, Polyomino
Wikipedia, The 35 hexominoes
Wikipedia, The 108 heptominoes
Wikipedia, The 369 octominoes
Wikipedia, Polyomino
D. Xu, T. Horiyama, T. Shirakawa and R. Uehara, Common Developments of Three Incongruent Boxes of Area 30, in Proc. 12th Annual Conference, TAMC 2015, Singapore, May 18-20, 2015, LNCS Vol. 9076, pp. 236-247.
L. Zucca, Pentominoes
L. Zucca, The 12 pentominoes
Index entries for "core" sequences

Sequences classifying polyominoes by symmetry group: A006746, A006747, A006748, A006749, A056877, A056878, A142886, A144553, A144554.
Cf. A001168 (not reduced by D_8 symmetry), A000104 (no holes), A054359, A054360, A001419, A000988, A030228 (chiral polyominoes).
See A006765 for another version.
Cf. also A000577, A000228, A103465, A210996 (bisection).
Excluding a(0), 8th and 9th row of A366766.

(* In this program by Jaime Rangel-Mondragón, polyominoes are represented as a list of Gaussian integers. *)
polyominoQ[p_List] := And @@ ((IntegerQ[Re[#]] && IntegerQ[Im[#]])& /@ p);
rot[p_?polyominoQ] := I*p;
ref[p_?polyominoQ] := (# - 2 Re[#])& /@ p;
cyclic[p_] := Module[{i = p, ans = {p}}, While[(i = rot[i]) != p, AppendTo[ans, i]]; ans];
dihedral[p_?polyominoQ] := Flatten[{#, ref[#]}& /@ cyclic[p], 1];
canonical[p_?polyominoQ] := Union[(# - (Min[Re[p]] + Min[Im[p]]*I))& /@ p];
allPieces[p_] := Union[canonical /@ dihedral[p]];
polyominoes[1] = {{0}};
polyominoes[n_] := polyominoes[n] = Module[{f, fig, ans = {}}, fig = ((f = #1; ({f, #1 + 1, f, #1 + I, f, #1 - 1, f, #1 - I}&) /@ f)&) /@ polyominoes[n - 1]; fig = Partition[Flatten[fig], n]; f = Select[Union[canonical /@ fig], Length[#1] == n &]; While[f != {}, ans = {ans, First[f]}; f = Complement[f, allPieces[First[f]]]]; Partition[Flatten[ans], n]];
a[n_] := a[n] = Length[ polyominoes[n]];
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 12}] (* Jean-François Alcover, Mar 24 2015, after Jaime Rangel-Mondragón *)

a(n) = A000104(n) + A001419(n). - R. J. Mathar, Jun 15 2014
a(n) = A006749(n) + A006746(n) + A006748(n) + A006747(n) + A056877(n) + A056878(n) + A144553(n) + A142886(n). - Andrew Howroyd, Dec 04 2018
a(n) = A259087(n) + A259088(n). - R. J. Mathar, May 22 2019
a(n) = (4*A006746(n) + 4*A006748(n) + 4*A006747(n) + 6*A056877(n) + 6*A056878(n) + 6*A144553(n) + 7*A142886(n) + A001168(n))/8. - John Mason, Nov 14 2021

Extended to n=28 by Tomás Oliveira e Silva
Link updated by William Rex Marshall, Dec 16 2009
Edited by Gill Barequet, May 24 2011
Misspelling "polyominos" corrected by Don Knuth, May 03 2016
a(29)-a(45), a(47) from Toshihiro Shirakawa
a(46) calculated using values from A001168 (I. Jensen), A006748/A056877/A056878/A144553/A142886 (Robert A. Russell) and A006746/A006747 (John Mason), Nov 14 2021

A000228 Number of hexagonal polyominoes (or hexagonal polyforms, or planar polyhexes) with n cells.

Original entry on oeis.org

1, 1, 3, 7, 22, 82, 333, 1448, 6572, 30490, 143552, 683101, 3274826, 15796897, 76581875, 372868101, 1822236628, 8934910362, 43939164263, 216651036012, 1070793308942, 5303855973849, 26323064063884, 130878392115834, 651812979669234, 3251215493161062, 16240020734253127, 81227147768301723, 406770970805865187, 2039375198751047333

Offset: 1

N. J. A. Sloane

From Markus Voege, Nov 24 2009: (Start)
On the difference between this sequence and A038147:
The first term that differs is for n=6; for all subsequent terms, the number of polyhexes is larger than the number of planar polyhexes.
If I recall correctly, polyhexes are clusters of regular hexagons that are joined at the edges and are LOCALLY embeddable in the hexagonal lattice.
"Planar polyhexes" are polyhexes that are GLOBALLY embeddable in the honeycomb lattice.
Example: (Planar) polyhex with 6 cells (x) and a hole (O):
.. x x
. x O x
.. x x
Polyhex with 6 cells that is cut open (I):
.. xIx
. x O x
.. x x
This polyhex is not globally embeddable in the honeycomb lattice, since adjacent cells of the lattice must be joined. But it can be embedded locally everywhere. It is a start of a spiral. For n>6 the spiral can be continued so that the cells overlap.
Illegal configuration with cut (I):
.. xIx
. x x x
.. x x
This configuration is NOT a polyhex since the vertex at
.. xIx
... x
is not embeddable in the honeycomb lattice.
One has to keep in mind that these definitions are inspired by chemistry. Hence, potential molecules are often the motivation for these definitions. Think of benzene rings that are fused at a C-C bond.
The (planar) polyhexes are "free" configurations, in contrast to "fixed" configurations as in A001207 = Number of fixed hexagonal polyominoes with n cells.
A000228 (planar polyhexes) and A001207 (fixed hexagonal polyominoes) differ only by the attribute "free" vs. "fixed," that is, whether the different orientations and reflections of an embedding in the lattice are counted.
The configuration
. x x .... x
.. x .... x x
is counted once as free and twice as fixed configurations.
Since most configurations have no symmetry, (A001207 / A000228) -> 12 for n -> infinity. (End)

A. T. Balaban and F. Harary, Chemical graphs V: enumeration and proposed nomenclature of benzenoid cata-condensed polycyclic aromatic hydrocarbons, Tetrahedron 24 (1968), 2505-2516.
A. T. Balaban and Paul von R. Schleyer, "Graph theoretical enumeration of polymantanes", Tetrahedron, (1978), vol. 34, 3599-3609
M. Gardner, Polyhexes and Polyaboloes. Ch. 11 in Mathematical Magic Show. New York: Vintage, pp. 146-159, 1978.
M. Gardner, Tiling with Polyominoes, Polyiamonds and Polyhexes. Chap. 14 in Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, pp. 175-187, 1988.
J. V. Knop et al., On the total number of polyhexes, Match, No. 16 (1984), 119-134.
W. F. Lunnon, Counting hexagonal and triangular polyominoes, pp. 87-100 of R. C. Read, editor, Graph Theory and Computing. Academic Press, NY, 1972.
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).

John Mason and Robert A. Russell, Table of n, a(n) for n = 1..36
Frédéric Chyzak, Ivan Gutman, and Peter Paule, Predicting the number of hexagonal systems with 24 and 25 hexagons, Communications in Mathematical and Computer Chemistry (1999) No. 40, 139-151. See p. 141.
A. Clarke, Polyhexes
F. Harary and R. C. Read, The enumeration of tree-like polyhexes, Proc. Edinburgh Math. Soc. (2) 17 (1970), 1-13.
D. Gouyou-Beauchamps and P. Leroux, Enumeration of symmetry classes of convex polyominoes on the honeycomb lattice, arXiv:math/0403168 [math.CO], 2004.
M. Keller, Counting polyforms
D. A. Klarner, Cell growth problems, Canad. J. Math. 19 (1967) 851-863.
J. V. Knop, K. Szymanski, Ž. Jeričević, and N. Trinajstić, On the total number of polyhexes, Match, No. 16 (1984), 119-134.
Greg Malen, Érika Roldán, and Rosemberg Toalá-Enríquez, Extremal {p, q}-Animals, Ann. Comb. (2023), p. 3.
John Mason, Counting polyhexes of size 36, updated Oct 27 2023.
Joseph Myers, Polyomino, polyhex and polyiamond tiling
Ed Pegg, Jr., Illustrations of polyforms
Jaime Rangel-Mondragon, Polyominoes and Related Families, The Mathematica Journal, 9:3 (2005), 609-640.
N. J. A. Sloane, Illustration of initial terms
N. Trinajstich, Z. Jerievi, J. V. Knop, W. R. Muller and K. Szymanski, Computer Generation of Isomeric Structures, Pure & Appl. Chem., Vol. 55, No. 2, pp. 379-390, 1983.
Eric Weisstein's World of Mathematics, Polyhex.

Equals (A006535 + A030225)/2.
Cf. A036359, A002216, A005963, A000228, A001998, A018190.
Cf. A001207, A057973, A364306.
Cf. also A000105, A000577, A103465, A057779, A258206, A258019.

a(13) from Achim Flammenkamp, Feb 15 1999
a(14) from Brendan Owen, Dec 31 2001
a(15) from Joseph Myers, May 05 2002
a(16)-a(20) from Joseph Myers, Sep 21 2002
a(21) from Herman Jamke (hermanjamke(AT)fastmail.fm), May 05 2007
a(22)-a(30) from John Mason, Jul 18 2023

A000207 Number of inequivalent ways of dissecting a regular (n+2)-gon into n triangles by n-1 non-intersecting diagonals under rotations and reflections; also the number of (unlabeled) maximal outerplanar graphs on n+2 vertices.

Original entry on oeis.org

1, 1, 1, 3, 4, 12, 27, 82, 228, 733, 2282, 7528, 24834, 83898, 285357, 983244, 3412420, 11944614, 42080170, 149197152, 531883768, 1905930975, 6861221666, 24806004996, 90036148954, 327989004892, 1198854697588, 4395801203290, 16165198379984, 59609171366326, 220373278174641

Offset: 1

N. J. A. Sloane

Also a(n) is the number of hexaflexagons of order n+2. - Mike Godfrey (m.godfrey(AT)umist.ac.uk), Feb 25 2002 (see the Kosters paper).
Number of normally non-isomorphic realizations of the associahedron of type II with dimension n in Ceballos et al. - Tom Copeland, Oct 19 2011
Number of polyforms with n cells in the hyperbolic tiling with Schläfli symbol {3,oo}, not distinguishing enantiomorphs. - Thomas Anton, Jan 16 2019
A stereographic projection of the {3,oo} tiling on the Poincaré disk can be obtained via the Christensson link. - Robert A. Russell, Jan 20 2024
A maximal outerplanar graph (MOP) has a plane embedding with all vertices on the exterior region and interior regions triangles. - Allan Bickle, Feb 25 2024

			E.g., a square (4-gon, n=2) could have either diagonal drawn, C(3)=2, but with essentially only one result. A pentagon (5-gon, n=3) gives C(4)=5, but they each have 2 diags emanating from 1 of the 5 vertices and are essentially the same. A hexagon can have a nuclear disarmament sign (6 ways), an N (3 ways and 3 reflections) or a triangle (2 ways) of diagonals, 6 + 6 + 2 = 14 = C(5), but only 3 essentially different. - _R. K. Guy_, Mar 06 2004
G.f. = x + x^2 + x^3 + 3*x^4 + 4*x^5 + 12*x^6 + 27*x^7 + 82*x^8 + ...

L. W. Beineke and R. E. Pippert, Enumerating labeled k-dimensional trees and ball dissections, pp. 12-26 of Proceedings of Second Chapel Hill Conference on Combinatorial Mathematics and Its Applications, University of North Carolina, Chapel Hill, 1970. Reprinted in Math. Annalen, 191 (1971), 87-98.
Cameron, Peter J. Some treelike objects. Quart. J. Math. Oxford Ser. (2) 38 (1987), no. 150, 155--183. MR0891613 (89a:05009). See pp. 155, 163, but note that the formulas on p. 163, lines 5 and 6, contain typos. See the correct formulas given here. - N. J. A. Sloane, Apr 18 2014
B. N. Cyvin, E. Brendsdal, J. Brunvoll and S. J. Cyvin, Isomers of polyenes attached to benzene, Croatica Chemica Acta, 68 (1995), 63-73.
S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751.
C. F. Earl and L. J. March, Architectural applications of graph theory, pp. 327-355 of R. J. Wilson and L. W. Beineke, editors, Applications of Graph Theory. Academic Press, NY, 1979.
R. K. Guy, "Dissecting a polygon into triangles," Bull. Malayan Math. Soc., Vol. 5, pp. 57-60, 1958.
R. K. Guy, Dissecting a polygon into triangles, Research Paper #9, Math. Dept., Univ. Calgary, 1967.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 79, Table 3.5.1 (the entries for n=16 and n=21 appear to be incorrect).
M. Kosters, A theory of hexaflexagons, Nieuw Archief Wisk., 17 (1999), 349-362.
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. K. Stockmeyer, The charm bracelet problem and its applications, pp. 339-349 of Graphs and Combinatorics (Washington, Jun 1973), Ed. by R. A. Bari and F. Harary. Lect. Notes Math., Vol. 406. Springer-Verlag, 1974.

T. D. Noe, Table of n, a(n) for n = 1..200
F. R. Bernhart & N. J. A. Sloane, Correspondence, 1977
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
Douglas Bowman and Alon Regev, Counting symmetry classes of dissections of a convex regular polygon, arXiv preprint arXiv:1209.6270 [math.CO], 2012.
William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
P. J. Cameron, Some treelike objects, Quart. J. Math. Oxford, 38 (1987), 155-183. See p. 160.
C. Ceballos, F. Santos, and G. Ziegler, Many Non-equivalent Realizations of the Associahedron, arXiv:1109.5544 [math.MG], 2011-2013, p. 19 and 26.
Malin Christensson, Make hyperbolic tilings of images, web page, 2019.
Sean Cleary, Roland Maio, Counting difficult tree pairs with respect to the rotation distance problem, arXiv:2001.06407 [cs.DS], 2020.
A. S. Conrad and D. K. Hartline, Flexagons
S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751. [Annotated scanned copy]
R. K. Guy, Dissecting a polygon into triangles, Research Paper #9, Math. Dept., Univ. Calgary, 1967. [Annotated scanned copy]
F. Harary and E. M. Palmer, On acyclic simplicial complexes, Mathematika 15 1968 115-122.
F. Harary, E. M. Palmer, and R. C. Read, On the cell-growth problem for arbitrary polygons, computer printout, circa 1974
F. Harary, E. M. Palmer and R. C. Read, On the cell-growth problem for arbitrary polygons, Discr. Math. 11 (1975), 371-389 (the entries for n=4 and n=30 appear to be incorrect).
J. W. Moon and L. Moser, Triangular dissections of n-gons, Canad. Math. Bull., 6 (1963), 175-178.
T. Motzkin, The hypersurface cross ratio, Bull. Amer. Math. Soc., 51 (1945), 976-984.
T. S. Motzkin, Relations between hypersurface cross ratios and a combinatorial formula for partitions of a polygon, for permanent preponderance and for non-associative products, Bull. Amer. Math. Soc., 54 (1948), 352-360 (the entry for n=10 appears to be incorrect).
C. O. Oakley and R. J. Wisner, Flexagons, Amer. Math. Monthly 64 (1957), 143-154.
Hans Rademacher, On the number of certain types of polyhedra, Illinois Journal of Mathematics 9.3 (1965): 361-380. Reprinted in Coll. Papers, Vol II, MIT Press, 1974, pp. 544-564.
Manfred Scheucher, Hendrik Schrezenmaier, Raphael Steiner, A Note On Universal Point Sets for Planar Graphs, arXiv:1811.06482 [math.CO], 2018.
Len Smiley, Illustration of initial terms
Tiberiu Spircu and Stefan V. Pantazi, Again around frieze patterns, arXiv:2002.08211 [math.CO], 2020. See Kn p. 13.
P. J. Stockmeyer, The charm bracelet problem and its applications, pp. 339-349 of Graphs and Combinatorics (Washington, Jun 1973), Ed. by R. A. Bari and F. Harary. Lect. Notes Math., Vol. 406. Springer-Verlag, 1974. [Scanned annotated and corrected copy]

Column k=3 of A295260.
Cf. A000577, A070765.
A row or column of the array in A169808.
Polyominoes: A001683(n+2) (oriented), A369314 (chiral), A208355(n-1) (achiral), A005036 {4,oo}, A007173 {3,3,oo}.
Cf. A097998, A097999, A098000 (labeled outerplanar graphs).
Cf. A111563, A111564, A111758, A111759, A111757 (unlabeled outerplanar graphs).

A000108 := proc(n) if n >= 0 then binomial(2*n,n)/(n+1) ; else 0; fi; end:
A000207 := proc(n) option remember: local k, it1, it2;
if n mod 2 = 0 then k := n/2+2 else k := (n+3)/2 fi:
if n mod 2 <> 0 then it1 := 0 else it1 := 1 fi:
if (n+2) mod 3 <> 0 then it2 := 0 else it2 := 1 fi:
RETURN(A000108(n)/(2*n+4) + it1*A000108(n/2)/4 + A000108(k-2)/2 + it2*A000108((n-1)/3)/3)
end:
seq(A000207(n),n=1..30) ; # (Revised Maple program from R. J. Mathar, Apr 19 2009)
A000207 := proc(n) option remember: local k,it1,it2; if n mod 2 = 0 then k := n/2+1 else k := (n+1)/2 fi: if n mod 2 <> 0 then it1 := 0 else it1 := 1 fi: if n mod 3 <> 0 then it2 := 0 else it2 := 1 fi: RETURN(A000108(n-2)/(2*n) + it1*A000108(n/2+1-2)/4 + A000108(k-2)/2 + it2*A000108(n/3+1-2)/3) end:
A000207 := n->(A000108(n)/(n+2)+A000108(floor(n/2))*((1+(n+1 mod 2) /2)))/2+`if`(n mod 3=1,A000108(floor((n-1)/3))/3,0); # Peter Luschny, Apr 19 2009 and M. F. Hasler, Apr 19 2009
G:=(12*(1+x-2*x^2)+(1-4*x)^(3/2)-3*(3+2*x)*(1-4*x^2)^(1/2)-4*(1-4*x^3)^(1/2))/24/x^2: Gser:=series(G,x=0,35): seq(coeff(Gser,x^n),n=1..31); # Emeric Deutsch, Dec 19 2004

p=3; Table[(Binomial[(p-1)n, n]/(((p-2)n+1)((p-2)n+2)) + If[OddQ[n], If[OddQ[p], Binomial[(p-1)n/2, (n-1)/2]/n, (p+1)Binomial[((p-1)n-1)/2, (n-1)/2]/((p-2)n+2)], 3Binomial[(p-1)n/2, n/2]/((p-2)n+2)]+Plus @@ Map[EulerPhi[ # ]Binomial[((p-1)n+1)/#, (n-1)/# ]/((p-1)n+1)&, Complement[Divisors[GCD[p, n-1]], {1, 2}]])/2, {n, 1, 20}] (* Robert A. Russell, Dec 11 2004 *)
a[n_] := (CatalanNumber[n]/(n+2) + CatalanNumber[ Quotient[n, 2]] *((1 + Mod[n-1, 2]/2)))/2 + If[Mod[n, 3] == 1, CatalanNumber[ Quotient[n-1, 3]]/3, 0] ; Table[a[n], {n, 1, 28}] (* Jean-François Alcover, Sep 08 2011, after PARI *)

A000207(n)=(A000108(n)/(n+2)+A000108(n\2)*if(n%2,1,3/2))/2+if(n%3==1,A000108(n\3)/3) \\ M. F. Hasler, Apr 19 2009

a(n) = C(n)/(2*n) + C(n/2+1)/4 + C(k)/2 + C(n/3+1)/3 where C(n) = A000108(n-2) if n is an integer, 0 otherwise and k = (n+1)/2 if n is odd, k = n/2+1 if n is even. Thus C(2), C(3), C(4), C(5), ... are 1, 1, 2, 5, ...
G.f.: (12*(1+x-2*x^2) + (1-4*x)^(3/2) - 3*(3+2*x)*(1-4*x^2)^(1/2) - 4*(1-4*x^3)^(1/2))/(24*x^2). - Emeric Deutsch, Dec 19 2004, from the S. J. Cyvin et al. reference.
a(n) ~ A000108(n)/(2*n+4) ~ 4^n / (2 sqrt(n Pi)*(n + 1)*(n + 2)). - M. F. Hasler, Apr 19 2009
a(n) = A001683(n+2) - A369314(n) = (A001683(n+2) + A208355(n-1)) / 2 = A369314(n) + A208355(n-1). - Robert A. Russell, Jan 19 2024
Beineke and Pippert have an explicit formula with six cases (based on the value of n mod 6). - Allan Bickle, Feb 25 2024

More terms from James Sellers, Jul 10 2000

A343909 Number of generalized polyforms on the tetrahedral-octahedral honeycomb with n cells.

Original entry on oeis.org

1, 2, 1, 4, 9, 44, 195, 1186, 7385, 49444, 337504, 2353664, 16608401, 118432965, 851396696, 6163949361, 44896941979

Offset: 0

Drake Thomas and Peter Kagey, May 03 2021

This sequence counts "free" polyforms where holes are allowed. This means that two polyforms are considered the same if one is a rigid transformation (translation, rotation, reflection, or a combination thereof) of the other.

			For n = 1, the a(1) = 2 polyforms are the tetrahedron and the octahedron.
For n = 2, the a(2) = 1 polyform is a tetrahedron and an octahedron connected at a face.
For n = 3, there are a(3) = 4 polyforms with 3 cells:
  - 3 consisting of one octahedron with two tetrahedra, and
  - 1 consisting of two octahedra and one tetrahedron.
For n = 4, there are a(4) = 9 polyforms with 4 cells:
  - 3 with one octahedron and three tetrahedra,
  - 5 with two octahedra and three octahedra, and
  - 1 with three octahedra and one tetrahedron.
For n = 5, there are a(5) = 44 polyforms with 5 cells:
  - 6 with one octahedron and four tetrahedra,
  - 24 with two octahedra and three tetrahedra,
  - 13 with three octahedra and two tetrahedra, and
  - 1 with four octahedra and one tetrahedron.

Peter Kagey, Animation of the a(4) = 9 polyforms with 4 cells.
Peter Kagey, Octahedron to tetrahedron ratio in generalized polyominoes in the tetrahedral-octahedral honeycomb, Mathematics Stack Exchange.
Wikipedia, Tetrahedral-octahedral honeycomb

Row sums of A365970.

Analogous for other honeycombs/tilings: A000105 (square), A000228 (hexagonal), A000577 (triangular), A038119 (cubical), A068870 (tesseractic), A197156 (prismatic pentagonal), A197159 (floret pentagonal), A197459 (rhombille), A197462 (kisrhombille), A197465 (tetrakis square), A309159 (snub square), A343398 (trihexagonal), A343406 (truncated hexagonal), A343577 (truncated square).

a(11)-a(16) from Bert Dobbelaere, Jun 10 2025

A197465 Number of free tetrakis polyaboloes (poly-[4.8^2]-tiles) with n cells, allowing holes, where division into tetrakis cells (triangular quarters of square grid cells) is significant.

Original entry on oeis.org

1, 2, 2, 6, 8, 22, 42, 112, 252, 650, 1584, 4091, 10369, 26938, 69651, 182116, 476272, 1253067, 3302187, 8733551, 23142116, 61477564, 163612714, 436278921, 1165218495, 3117021788, 8349892686, 22397754046, 60153261611

Offset: 1

Joseph Myers, Oct 15 2011

See the link below for a definition of the tetrakis square tiling. When a square grid cell is divided into triangles, it must be divided dexter (\) or sinister (/) according to the parity of the grid cell.

			For n=3 there are 4 triaboloes.  Of these, 2 conform to the tetrakis grid.  Each of these 2 has a unique dissection into 6 tetrakis cells. - _George Sicherman_, Mar 25 2021

Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, Sections 2.7, 6.2 and 9.4.

Peter Kagey, Example illustrating a(4) = 6.
Wikipedia, Tetrakis square tiling

Cf. A197466, A197467, A006074.

Analogous for other tilings: A000105 (square), A000228 (hexagonal), A000577 (triangular), A197156 (prismatic pentagonal), A197159 (floret pentagonal), A197459 (rhombille), A197462 (kisrhombille), A309159 (snub square), A343398 (trihexagonal), A343406 (truncated hexagonal), A343577 (truncated square).

Name clarified by George Sicherman, Mar 25 2021
a(21)-a(26) from Aaron N. Siegel, May 18 2022
a(27)-a(29) from Bert Dobbelaere, Jun 04 2025

A343577 Number of generalized polyforms on the truncated square tiling with n cells.

Original entry on oeis.org

1, 2, 2, 7, 22, 93, 413, 2073, 10741, 57540, 312805, 1722483, 9564565, 53489304, 300840332, 1700347858, 9650975401

Offset: 0

Peter Kagey, Apr 20 2021

a(n) >= A343417(n), the number of (n-k)-polyominoes with k distinguished vertices.

Kadon Enterprises, Interesting solutions: Ochominoes.
Peter Kagey, Haskell program for computing sequence.
Peter Kagey, The a(3) = 7 generalized polyforms on the truncated square tiling with 3 cells.
Peter Kagey, The a(5) = 93 generalized polyforms on the truncated square tiling with 5 cells.
John Mason, Illustration of equivalence between truncated square polyforms and a mixture of crosses and squares.
Wikipedia, Truncated Square Tiling

Cf. A121197 (one-sided).

Analogous for other tilings: A000105 (square), A000228 (hexagonal), A000577 (triangular), A197156 (prismatic pentagonal), A197159 (floret pentagonal), A197459 (rhombille), A197462 (kisrhombille), A197465 (tetrakis square), A309159 (snub square), A343398 (trihexagonal), A343406 (truncated hexagonal).

a(11) from Drake Thomas, May 02 2021

a(12)-a(16) from John Mason, Mar 20 2022

A001420 Number of fixed 2-dimensional triangular-celled animals with n cells (n-iamonds, polyiamonds) in the 2-dimensional hexagonal lattice.

Original entry on oeis.org

2, 3, 6, 14, 36, 94, 250, 675, 1838, 5053, 14016, 39169, 110194, 311751, 886160, 2529260, 7244862, 20818498, 59994514, 173338962, 501994070, 1456891547, 4236446214, 12341035217, 36009329450, 105229462401, 307942754342, 902338712971, 2647263986022, 7775314024683, 22861250676074, 67284446545605

Offset: 1

N. J. A. Sloane

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.

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

Vaclav Kotesovec, Table of n, a(n) for n = 1..75 (from reference by A. J. Guttmann)
G. Aleksandrowicz and G. Barequet, counting d-dimensional polycubes and nonrectangular planar polyomnoes, Lect. Not. Comp. Sci 4112 (2006) 418-427 Table 3
G. Aleksandrowicz and G. Barequet, Counting d-dimensional polycubes and nonrectangular planar polyominoes, Int. J. of Computational Geometry and Applications, 19 (2009), 215-229.
Gill Barequet, Solomon W. Golomb, and David A. Klarner, Polyominoes. (This is a revision, by G. Barequet, of the chapter of the same title originally written by the late D. A. Klarner for the first edition, and revised by the late S. W. Golomb for the second edition.) Preprint, 2016.
Gill Barequet and Mirah Shalah, Improved Bounds on the Growth Constant of Polyiamonds, 32nd European Workshop on Computational Geometry, 2016.
Gill Barequet, Mira Shalah, and Yufei Zheng, An Improved Lower Bound on the Growth Constant of Polyiamonds, In: Cao Y., Chen J. (eds) Computing and Combinatorics, COCOON 2017, Lecture Notes in Computer Science, vol 10392.
Vuong Bui, The number of polyiamonds is almost supermultiplicative, arXiv:2304.10077 [math.CO], 2023.
A. J. Guttmann (ed.), Polygons, Polyominoes and Polycubes, Lecture Notes in Physics, 775 (2009). (Table 16.11, p. 479 has 75 terms of this sequence.)
Greg Malen, Érika Roldán, and Rosemberg Toalá-Enríquez, Extremal {p, q}-Animals, Ann. Comb. (2023), p. 3.
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
H. Redelmeier, Emails to N. J. A. Sloane, 1991

Cf. A000577, A001168, A006534, A030223, A030224.

More terms from Brendan Owen (brendan_owen(AT)yahoo.com), Dec 15 2001
a(28) from Joseph Myers, Sep 24 2002
a(29)-a(31) from the Aleksandrowicz and Barequet paper (N. J. A. Sloane, Jul 09 2009)
Slightly edited by Gill Barequet, May 24 2011
a(32) from Paul Church, Oct 06 2011

cp's OEIS Frontend

A057729 Number of triangular polyominoes (or polyiamonds) [A000577] with perimeter n.

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A000105 Number of free polyominoes (or square animals) with n cells.

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A000228 Number of hexagonal polyominoes (or hexagonal polyforms, or planar polyhexes) with n cells.

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A000207 Number of inequivalent ways of dissecting a regular (n+2)-gon into n triangles by n-1 non-intersecting diagonals under rotations and reflections; also the number of (unlabeled) maximal outerplanar graphs on n+2 vertices.

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A343909 Number of generalized polyforms on the tetrahedral-octahedral honeycomb with n cells.

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A197465 Number of free tetrakis polyaboloes (poly-[4.8^2]-tiles) with n cells, allowing holes, where division into tetrakis cells (triangular quarters of square grid cells) is significant.

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A343577 Number of generalized polyforms on the truncated square tiling with n cells.

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A001420 Number of fixed 2-dimensional triangular-celled animals with n cells (n-iamonds, polyiamonds) in the 2-dimensional hexagonal lattice.

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