Displaying 1-10 of 12 results found.
Number of fixed 2-dimensional triangular-celled animals with n cells (n-iamonds, polyiamonds) in the 2-dimensional hexagonal lattice.
(Formerly M0806 N0305)
+0
12
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
COMMENTS
The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
REFERENCES
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).
LINKS
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.
EXTENSIONS
More terms from Brendan Owen (brendan_owen(AT)yahoo.com), Dec 15 2001
a(29)-a(31) from the Aleksandrowicz and Barequet paper ( N. J. A. Sloane, Jul 09 2009)
Number of fixed hexagonal polyominoes with n cells.
(Formerly M2897 N1162)
+0
20
1, 3, 11, 44, 186, 814, 3652, 16689, 77359, 362671, 1716033, 8182213, 39267086, 189492795, 918837374, 4474080844, 21866153748, 107217298977, 527266673134, 2599804551168, 12849503756579, 63646233127758, 315876691291677, 1570540515980274, 7821755377244303, 39014584984477092, 194880246951838595, 974725768600891269, 4881251640514912341, 24472502362094874818, 122826412768568196148, 617080993446201431307, 3103152024451536273288, 15618892303340118758816, 78679501136505611375745
REFERENCES
A. J. Guttmann, ed., Polygons, Polyominoes and Polycubes, Springer, 2009, p. 477. (Table 16.9 has 46 terms of this sequence.)
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).
LINKS
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.
EXTENSIONS
More terms from Markus Voege (markus.voege(AT)inria.fr), Mar 25 2004
a(n) is the number of fixed polyglasses (polyiamonds which need only touch at corners) with n cells.
+0
4
2, 12, 88, 710, 6054, 53500, 484784, 4475010, 41902626, 396838992, 3793117200, 36534684066
COMMENTS
Polyglasses are to polyiamonds ( A001420) as polyplets ( A006770) are to polyominoes ( A001168). The name derives from the 2-celled animal (diglass) which looks like an hourglass.
EXAMPLE
a(2) = 12: three rotations of a diamond, three rotations of an hourglass and six rotations of "two mountains".
Number of fixed 3-dimensional polycubes with n cells; lattice animals in the simple cubic lattice (6 nearest neighbors), face-connected cubes.
(Formerly M2996 N1213)
+0
19
1, 3, 15, 86, 534, 3481, 23502, 162913, 1152870, 8294738, 60494549, 446205905, 3322769321, 24946773111, 188625900446, 1435074454755, 10977812452428, 84384157287999, 651459315795897, 5049008190434659, 39269513463794006, 306405169166373418
COMMENTS
This gives the number of polycubes up to translation (but not rotation or reflection). - Charles R Greathouse IV, Oct 08 2013
REFERENCES
W. F. Lunnon, Symmetry of cubical and general polyominoes, pp. 101-108 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).
LINKS
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.
EXTENSIONS
a(18) from the Aleksandrowicz and Barequet paper ( N. J. A. Sloane, Jul 09 2009)
Number of fixed n-polyominoids, allowing both corner- and edge-connections.
+0
4
3, 48, 1072, 27732, 781200
Number of n-polyplets (polyominoes connected at edges or corners); may contain holes.
+0
20
1, 2, 5, 22, 94, 524, 3031, 18770, 118133, 758381, 4915652, 32149296, 211637205, 1401194463, 9321454604, 62272330564, 417546684096
COMMENTS
See A056840 for illustrations, valid also for this sequence up to n=4, but slightly misleading for polyplets with holes. See the colored areas in the illustration of A056840(5)=99 which correspond to identical 5-polyplets. (The 2+2+4-3 = 5 additional figures counted there correspond to the 4-square configuration with a hole inside ({2,4,6,8} on a numeric keyboard), with one additional square added in three inequivalent places: "inside" one angle (touching two sides), attached to one side, and attached to a corner. These do only count for 3 here, but for 8 in A056840.) It can be seen that A056840 counts a sort of "spanning trees" instead, i.e., simply connected graphs that connect all of the vertices (using only "King's moves", and maybe other additional constraints). - M. F. Hasler, Sep 29 2014
LINKS
Eric Weisstein's World of Mathematics, Polyplet.
EXAMPLE
XXX..XX...XX..X.X..X.. (the 5 for n=3)
.......X...X...X....X.
.....................X
Number of free polyominoids with n cells, allowing flat corner-connections and right-angled edge-connections.
+0
17
COMMENTS
This sequence and the related sequences A365650- A365655 and A365996- A366010 count polyominoids ( A075679) with different rules for how the cells can be connected. In these sequences, connections other than the specified ones are permitted, but the polyominoids must be connected through the specified connections only. The polyominoids counted by this sequence, for example, are allowed to have right-angled corner-connections and flat edge-connections, as long as they are not needed for the polyominoid to be connected. A connection is flat if the two neighboring cells lie in the same plane, otherwise it is right-angled.
CROSSREFS
The following table lists counting sequences for free, fixed, and one-sided polyominoids with different sets of allowed connections. "|" means flat connections and "L" means right-angled connections.
corner-connections | edge-connections | free | fixed | 1-sided
-------------------+------------------+---------+---------+--------
Array read by antidiagonals, where each row is the counting sequence of a certain type of fixed polyominoids.
+0
25
1, 0, 1, 0, 1, 2, 0, 1, 0, 2, 0, 1, 0, 2, 2, 0, 1, 0, 2, 4, 2, 0, 1, 0, 2, 12, 6, 1, 0, 1, 0, 2, 38, 22, 0, 1, 0, 1, 0, 2, 126, 88, 0, 2, 1, 0, 1, 0, 2, 432, 372, 0, 6, 2, 1, 0, 1, 0, 2, 1520, 1628, 0, 19, 6, 4, 3, 0, 1, 0, 2, 5450, 7312, 0, 63, 19, 20, 0, 3
COMMENTS
See A366766 (corresponding array for free polyominoids) for details.
EXAMPLE
Array begins:
n\k| 1 2 3 4 5 6 7 8 9 10 11 12
---+--------------------------------------------------------------------
1 | 1 0 0 0 0 0 0 0 0 0 0 0
2 | 1 1 1 1 1 1 1 1 1 1 1 1
3 | 2 0 0 0 0 0 0 0 0 0 0 0
4 | 2 2 2 2 2 2 2 2 2 2 2 2
5 | 2 4 12 38 126 432 1520 5450 19820 72892 270536 1011722
6 | 2 6 22 88 372 1628 7312 33466 155446 730534 3466170 16576874
7 | 1 0 0 0 0 0 0 0 0 0 0 0
8 | 1 2 6 19 63 216 760 2725 9910 36446 135268 505861
9 | 1 2 6 19 63 216 760 2725 9910 36446 135268 505861
10 | 1 4 20 110 638 3832 23592 147941 940982 6053180 39299408 257105146
11 | 3 0 0 0 0 0 0 0 0 0 0 0
12 | 3 3 3 3 3 3 3 3 3 3 3 3
CROSSREFS
The following table lists some sequences that are rows of the array, together with the corresponding values of D, d, and C (see A366766). Some sequences occur in more than one row. Notation used in the table:
X: Allowed connection.
-: Not allowed connection (but may occur "by accident" as long as it is not needed for connectedness).
.: Not applicable for (D,d) in this row.
!: d < D and all connections have h = 0, so these polyominoids live in d < D dimensions only.
*: Whether a connection of type (g,h) is allowed or not is independent of h.
| | | connections |
| | | g:112223 |
n | D | d | h:010120 | sequence
----+---+---+-------------+----------
34 | 3 | 3 | * X.X..- |
36 | 3 | 3 | * X.-..X |
37 | 3 | 3 | * -.X..X |
38 | 3 | 3 | * X.X..X |
Number of free polyplets with n cells that are not polyominoes.
+0
0
0, 1, 3, 17, 82, 489, 2923, 18401, 116848, 753726, 4898579, 32085696, 211398614, 1400292492, 9318028028, 62259251309, 417496576187
LINKS
Eric Weisstein's World of Mathematics, Polyplet.
EXAMPLE
XX...X.X..X.. (the 3 for n=3)
..X...X....X.
............X
a(3) = 5 - 2 = 3
a(4) = 22 - 5 = 17
a(5) = 94 - 12 = 82
a(n) is the number of d+/d- diagonally convex polyplets (polyominoes which need only touch at corners) with n cells.
+0
0
1, 4, 20, 108, 600, 3368, 18968, 106906, 602532, 3395402, 19131460, 107788900, 607274848
COMMENTS
A polyplet is d+ [d-] diagonally convex if the intersection of its interior with any line of slope 1 [-1] through the centers of the cells is connected.
EXAMPLE
The only tetraplets that are not d+ diagonally convex are two animals consisting of a horizontal domino and a vertical domino joined at a corner. So a(4) = A006770(4) - 2 = 108.
CROSSREFS
Cf. A006770 (fixed polyplets), A187077 (row convex polyplets), A187276 (d+/d- diagonally convex polyominoes).
Search completed in 0.010 seconds
|