Displaying 1-10 of 30 results found.
1, 2, 5, 12, 34, 115, 446, 1881, 8386, 38472, 179701, 849285, 4047541, 19415118, 93623028, 453486806, 2205081449, 10758731196, 52651373968, 258365785954, 1270930958357, 6265738653554, 30952863554094, 153191072337297
COMMENTS
Partial sums of number of planar simply-connected polyhexes (or benzenoid hydrocarbons) with n hexagons. The only known primes in the partial sum are 2 and 5.
EXAMPLE
a(6) = 1 + 1 + 3 + 7 + 22 + 81 = 115.
Number of hexagonal polyominoes (or hexagonal polyforms, or planar polyhexes) with n cells.
(Formerly M2682 N1072)
+10
73
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
COMMENTS
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)
REFERENCES
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).
LINKS
Eric Weisstein's World of Mathematics, Polyhex.
EXTENSIONS
a(14) from Brendan Owen, Dec 31 2001
a(21) from Herman Jamke (hermanjamke(AT)fastmail.fm), May 05 2007
Number of n-celled free polyominoes without holes.
(Formerly M1424 N0560)
+10
36
1, 1, 1, 2, 5, 12, 35, 107, 363, 1248, 4460, 16094, 58937, 217117, 805475, 3001127, 11230003, 42161529, 158781106, 599563893, 2269506062, 8609442688, 32725637373, 124621833354, 475368834568, 1816103345752, 6948228104703, 26618671505989, 102102788362303
REFERENCES
J. S. Madachy, Pentominoes - Some Solved and Unsolved Problems, J. Rec. Math., 2 (1969), 181-188.
George E. Martin, Polyominoes - A Guide to Puzzles and Problems in Tiling, The Mathematical Association of America, 1996
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
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)
CROSSREFS
Cf. A000105, row sums of A308300, A006746, A056877, A006748, A056878, A006747, A006749, A054361, A070765 (polyiamonds), A018190 (polyhexes), A266549 (by perimeter).
EXTENSIONS
Extended to n=26 by Tomás Oliveira e Silva
Number of strictly non-overlapping holeless polyhexes of perimeter 2n, counted up to rotations and turning over.
+10
16
0, 0, 1, 0, 1, 1, 3, 2, 12, 14, 50, 97, 312, 744, 2291, 6186, 18714, 53793, 162565, 482416, 1467094, 4436536, 13594266
COMMENTS
Differs from A057779 for the first time at n=12 as here a(12) = 97, one less than A057779(12) because this sequence excludes polyhexes with holes, the smallest which contains six hexagons in a ring, enclosing a hole of one hex, having thus perimeter of 18+6 = 24 (= 2*12) edges. Differs from A258019 for the first time at n=13 as here a(13) = 312, one less than A258019(13) because this sequence counts only strictly non-overlapping and non-touching polyhex-patterns, while A258019(13) already includes one specimen of helicene-like self-reaching structures. If one counts these structures by the number of hexagons (instead of perimeter length), one obtains sequence 1, 1, 3, 7, 22, 81, ... ( A018190). a(n) is also the number of 2n-step 2-dimensional closed self-avoiding paths on honeycomb lattice, reduced for symmetry. - Luca Petrone, Jan 08 2016
REFERENCES
S. J. Cyvin, J. Brunvoll and B. N. Cyvin, Theory of Coronoid Hydrocarbons, Springer-Verlag, 1991. See sections 4.7 Annulene and 6.5 Annulenes.
FORMULA
Other observations. For all n >= 1:
Number of polyhexes with n cells that tile the plane.
+10
12
1, 1, 3, 7, 22, 77, 294, 1054, 3788, 11326, 24790, 103641, 164559, 532510, 1574252, 2939898, 4761009, 21048218, 24306306, 95707819, 205176450
REFERENCES
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.
Number of planar cata-polyhexes with n cells.
+10
10
1, 1, 2, 5, 12, 36, 118, 411, 1489, 5572, 21115, 81121, 314075, 1224528, 4799205
COMMENTS
Number of cata-condensed benzenoid hydrocarbons with n hexagons.
a(n) is the number of n-celled polyhexes with perimeter 4n+2. 4n+2 is the maximal perimeter of an n-celled polyhex. a(n) is the number of n-celled polyhexes that have a tree as their connectedness graph (vertices of this graph correspond to cells and two vertices are connected if the corresponding cells have a common edge). - Tanya Khovanova, Jul 27 2007
REFERENCES
N. Trinajstić, S. Nikolić, J. V. Knop, W. R. Müller and K. Szymanski, Computational Chemical Graph Theory: Characterization, Enumeration, and Generation of Chemical Structures by Computer Methods, Ellis Horwood, 1991.
LINKS
Eric Weisstein's World of Mathematics, Polyhex. Eric Weisstein's World of Mathematics, Fusene.
Number of fusenes with n hexagons.
+10
9
1, 1, 3, 7, 22, 82, 339, 1505, 7036, 33836, 166246, 829987, 4197273, 21456444, 110716585, 576027737, 3018986040, 15927330105, 84530870455, 451069339063, 2418927725532, 13030938290472, 70492771581350, 382816374644336, 2086362209298079, 11408580755666756
LINKS
Eric Weisstein's World of Mathematics, Fusene
Number of polyhexes with n cells.
+10
8
1, 1, 3, 7, 22, 83, 341, 1519, 7114, 34350
CROSSREFS
See A000228 for another version of this sequence.
Polycyclic aromatic hydrocarbons (for precise definition see He and He, 1986).
+10
8
1, 1, 3, 7, 22, 81, 331, 1436, 6510, 30129, 141512, 671538, 3210620, 15443871, 74662005, 362506902
REFERENCES
N. Trinajstić, S. Nikolić, J. V. Knop, W. R. Müller and K. Szymanski, Computational Chemical Graph Theory: Characterization, Enumeration, and Generation of Chemical Structures by Computer Methods, Ellis Horwood, 1991. [incorrectly gives a(12) = 671512 in Table 4.13]
CROSSREFS
A018190 is a very similar sequence with (as He and He remark) a slightly different definition. Cf. A000228, A038147.
EXTENSIONS
a(10)-a(16) from Cyvin, Brunvoll & Cyvin (Table 1) added by Andrey Zabolotskiy, Feb 08 2023
Number of polyiamonds with n cells, without holes.
+10
7
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
COMMENTS
If holes are allowed, we get A000577.
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