A120102
Number of polyominoes consisting of 8 regular unit n-gons.
Original entry on oeis.org
66, 369, 2812, 1448, 2876, 10102, 34838, 73675, 181127, 131801, 185297, 352375, 725869, 1180526, 2104485, 1694978, 2123088, 3291481, 5402087, 7739008, 11832175, 10079003, 11917261, 16624712, 24389611, 32317393, 45260884
Offset: 3
a(3)=66 because there are 66 polyiamonds consisting of 8 triangles and a(4)=369 because there are 369 polyominoes consisting of 8 squares.
Cf.
A000577,
A000104,
A000228,
A103465,
A103466,
A103467,
A103468,
A103469,
A103470,
A103471,
A103472,
A103473,
A103465,
A120103,
A120104.
A120104
Number of polyominoes consisting of 10 regular unit n-gons.
Original entry on oeis.org
448, 4655, 76092, 30490, 80075, 430302, 2285047, 6078768, 20376032, 13303523, 21208739, 49734303, 131517548, 249598727, 540742895, 404616118, 549711709, 983715865, 1910489463, 3070327312
Offset: 3
a(3)=448 because there are 448 polyiamonds consisting of 10 triangles;
a(4)=4655 because there are 4655 polyominoes consisting of 10 squares.
A006534
Number of one-sided triangular polyominoes (n-iamonds) with n cells; turning over not allowed, holes are allowed.
Original entry on oeis.org
1, 1, 1, 4, 6, 19, 43, 120, 307, 866, 2336, 6588, 18373, 52119, 147700, 422016, 1207477, 3471067, 9999135, 28893560, 83665729, 242826187, 706074369, 2056870697, 6001555275, 17538335077, 51323792789, 150390053432, 441210664337, 1295886453860, 3810208448847, 11214076720061, 33035788241735
Offset: 1
From _M. F. Hasler_, Nov 12 2017: (Start)
Putting dots for the approximate center of the regular triangles (alternatively flipped up and down for neighboring dots), we have:
a(4) = #{ .... , .:. , ..: , :.. } = 4, while ..: and :.. are considered equivalent and not counted twice in A000577(4) = 3.
a(5) = #{ ..... , ...: , :... , ..:. , .:.. , :.: } = 6, and again the 2nd & 3rd and 4th & 5th are considered equivalent and not counted twice in A000577(5) = 4. (End)
- 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.
- 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.
- John Mason, Table of n, a(n) for n = 1..52
- 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.
- J. Meeus & N. J. A. Sloane, Correspondence, 1974-1975
- Ed Pegg, Jr., Illustrations of polyforms
- Eric Weisstein's World of Mathematics, Polyiamond.
A070764
Number of polyiamonds with n cells, with holes.
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 0, 0, 1, 4, 25, 108, 450, 1713, 6267, 21988, 75185, 251590, 828408, 2692630, 8661287, 27624040, 87479663, 275392248, 862593661, 2690285608, 8359581585, 25893044920, 79978118632, 246433568617
Offset: 1
A070765
Number of polyiamonds with n cells, without holes.
Original entry on oeis.org
1, 1, 1, 3, 4, 12, 24, 66, 159, 444, 1161, 3226, 8785, 24453, 67716, 189309, 528922, 1484738, 4172185, 11756354, 33174451, 93795220, 265565628, 753060469, 2138206966, 6078931114, 17302380313, 49302121747, 140627400927, 401510058179
Offset: 1
- Greg Huber, Craig Knecht, Walter Trump, and Robert M. Ziff, Riddles of the sphinx tilings, arXiv:2304.14388 [cond-mat.stat-mech], 2023.
- Elena V. Konstantinova and Maxim V. Vidyuk, Discriminating tests of information and topological indices. Animals and trees, J. Chem. Inf. Comput. Sci. 43 (2003), 1860-1871.
- Hai Pham-Van, Linh Tran-Phan-Thuy, Cuong Tran-Manh, Bich Do-Danh, and Hoang Luc-Huy, Two-Dimensional Clusters of Colloidal Particles Induced by Emulsion Droplet Evaporation, Nanomaterials (2020) Vol. 10, 156.
A103468
Number of polyominoes consisting of n regular unit 10-gons.
Original entry on oeis.org
1, 1, 4, 19, 127, 985, 8350, 73675, 664411, 6078768, 56198759, 523924389, 4918127659
Offset: 1
A120103
Number of polyominoes consisting of 9 regular unit n-gons.
Original entry on oeis.org
160, 1285, 14445, 6572, 14982, 65323, 280014, 664411, 1908239, 1314914, 1968684, 4158216, 9707046, 17054708, 33522023, 26019735, 33942901, 56537856, 100952307, 153177526, 251530341, 208524646, 254079408, 374310135, 586169115, 812395658
Offset: 3
a(3)=160 because there are 160 polyiamonds consisting of 9 triangles and a(4)=1285 because there are 1285 polyominoes consisting of 9 squares.
Cf.
A000577,
A000104,
A000105,
A000228,
A103465,
A103466,
A103467,
A103468,
A103469,
A103470,
A103471,
A103472,
A103473,
A103465,
A120102,
A120104.
A030224
Triangular polyominoes (n-iamonds) without bilateral symmetry (holes are allowed).
Original entry on oeis.org
0, 0, 0, 1, 2, 7, 19, 54, 147, 418, 1150, 3254, 9138, 25953, 73717, 210719, 603370, 1734739, 4998542, 14444576, 41829991, 121406927, 353029078, 1028417980, 3000754648, 8769118355, 25661830891, 75194886765, 220605144778, 647942827064, 1905103686527, 5607037213434, 16517892572160, 48703328674762
Offset: 1
A103466
Number of polyominoes consisting of n regular unit octagons.
Original entry on oeis.org
1, 1, 3, 11, 50, 269, 1605, 10102, 65323, 430302, 2868320, 19299334, 130807068, 892075515, 6115673262
Offset: 1
Cf.
A000577,
A000104,
A000228,
A103465,
A103467,
A103468,
A103469,
A103470,
A103471,
A103472,
A103473,
A115071,
A120102,
A120103,
A120104.
A103467
Number of polyominoes consisting of n regular unit 9-gons.
Original entry on oeis.org
1, 1, 3, 14, 82, 585, 4418, 34838, 280014, 2285047, 18838395, 156644526, 1311575691
Offset: 1
Cf.
A000577,
A000104,
A000228,
A103465,
A103466,
A103468,
A103469,
A103470,
A103471,
A103472,
A103473,
A115071,
A120102,
A120103,
A120104.
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