Displaying 1-10 of 14 results found.
1, 3, 4, 14, 30, 107, 318, 1106, 3671
REFERENCES
M. Gardner, Mathematical Magic Show. Random House, NY, 1978, p. 151.
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.
+10
14
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
COMMENTS
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.
REFERENCES
Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, Sections 2.7, 6.2 and 9.4.
EXAMPLE
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
CROSSREFS
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).
Number of convex polyaboloes (or convex polytans): number of distinct convex shapes that can be formed with n congruent isosceles right triangles. Reflections are not counted as different.
+10
8
1, 3, 2, 6, 3, 7, 5, 11, 5, 10, 7, 14, 7, 16, 11, 20, 9, 17, 13, 22, 12, 25, 18, 27, 14, 24, 20, 31, 18, 36, 26, 37, 19, 34, 28, 38, 24, 45, 34, 47, 26, 41, 36, 49, 35, 61, 44, 54, 32, 54, 45, 56, 40, 71, 56, 63, 40, 66, 56, 72, 49, 86, 66, 76, 51, 74, 67, 77
COMMENTS
Side numbers range from 3 to 8. See Wang and Hsiung (1942). - Douglas J. Durian, Sep 24 2017
LINKS
Paul Scott, Convex Tangrams, Australian Mathematics Teacher, 62 (2006), 2-5. Confirms a(16)=20. Fu Traing Wang and Chuan-Chih Hsiung, A Theorem on the Tangram, American Mathematical Monthly, 49 (1942), 596-599. Proves a(16)=20 and that convex polyabolos have no more than eight sides.
EXAMPLE
For n=3, there are two trapezoids.
CROSSREFS
Strictly less than A006074 for n > 2.
Number of free pseudo-polytans with n cells.
+10
4
1, 10, 91, 1432, 23547, 416177, 7544247, 139666895, 2623895224
COMMENTS
A pseudo-polytan is a planar figure consisting of n isosceles right triangles joined either edge-to-edge or corner-to-corner, in such a way that the short edges of the triangles coincide with edges of the square lattice. Two figures are considered equivalent if they differ only by a rotation or reflection.
The pseudo-polytans are constructed in the same way as ordinary polytans ( A006074), but allowing for corner-connections. Thus they generalize polytans in the same way that pseudo-polyominoes (aka polyplets, A030222) generalize ordinary polyominoes ( A000105).
EXAMPLE
a(2) = 10, because there are 10 ways of adjoining two isosceles right triangles: 3 distinct edge-to-edge joins (cf. A006074), and 7 distinct corner-to-corner joins.
Number of 1-sided polytans (polyaboloes) with n cells.
+10
3
1, 4, 6, 22, 56, 198, 624, 2182, 7448, 26319, 92826, 332181, 1192845, 4315845, 15678200, 57227380, 209623109, 770516966, 2840466846, 10499678185, 38905008340, 144475534207
LINKS
Eric Weisstein's World of Mathematics, Polytan
EXTENSIONS
a(15) corrected and a(16)-a(20) from John Mason, Jan 07 2022
Number of free pseudo-polyarcs with n cells.
+10
3
2, 32, 700, 21943, 737164, 25959013, 938559884
COMMENTS
See A057787 for a description of polyarcs. The pseudo-polyarcs are constructed in the same way as ordinary polyarcs, but allowing for corner-connections. Thus they generalize polyarcs in the same way that pseudo-polyominoes (aka polyplets, A030222) generalize ordinary polyominoes ( A000105). They can also be viewed as the "rounded" variant of pseudo-polytans ( A354380), in the same way that ordinary polyarcs are the rounded variant of ordinary polytans ( A006074). Two figures are considered equivalent if they differ only by a rotation or reflection.
The pseudo-polyarcs grow tremendously fast, much faster than most polyforms. The initial data that have been computed suggest an asymptotic growth rate of at least 36^n.
EXAMPLE
a(10) = 32, because there are 32 ways of adjoining two monarcs: 7 distinct edge-to-edge joins, and 25 distinct corner-to-corner joins (including one double-corner join involving two concave arcs).
Number of polyfetts (or polifetti) with n cells, identifying mirror images.
+10
2
1, 10, 90, 1414, 23136, 406093, 7303813, 134027098
COMMENTS
A polyfett is a generalized polyabolo (or polytan). Its cells are equal isosceles right triangles on the quadrille grid, which may be joined along equal edges or at vertices.
Polyfetts are to polyaboloes what polyplets (or polykings) are to polyominoes.
EXAMPLE
For n = 2, a(2) = 10. Two polyabolo cells can be joined at edges to form 3 different diaboloes, or at corners to form 7 different proper difetts.
PROG
(C) /* See link to Unix C program polyaboloes.c under LINKS. */
Number of 1-sided strip polytans with n cells.
+10
1
1, 4, 6, 21, 47, 134, 323, 876, 2224, 5885, 15146, 39574, 102250, 265574, 686208, 1775646, 4583718, 11835037, 30515520, 78652647
LINKS
Eric Weisstein's World of Mathematics, Polytan
Number of 2-sided strip polytans with n cells.
+10
1
1, 3, 4, 13, 25, 72, 166, 450, 1124, 2973, 7603, 19865, 51199, 132982, 343290, 888309, 2292325, 5918742, 15258937, 39329418
LINKS
Eric Weisstein's World of Mathematics, Polytan
Number of fixed polytans (polyaboloes) with n cells.
+10
1
4, 9, 24, 71, 224, 740, 2496, 8565, 29792, 104701, 371304, 1326702, 4771380, 17256161, 62712800, 228883359, 838492436, 3081972336, 11361867384, 41998361480, 155620033360, 577900838281
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