Demonstration of the
On-Line Encyclopedia of Integer Sequences
(Page 5)
Identifying a Sequence: a Sequence From a Chemical Journal
A chemist reading a technical journal comes
across the sequence
1, 3, 7, 22, 82, 333, 1448, ...
and wonders if there is a simple mathematical formula for it.
He or she goes to the main web page of the
On-Line Encyclopedia of Integer Sequences
and enters the sequence,
as in the previous demonstrations.
This produces the following response.
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| A000228 |
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Number of hexagonal polyominoes (or planar polyhexes) with n cells.
(Formerly M2682 N1072)
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+0
24
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| 1, 1, 3, 7, 22, 82, 333, 1448, 6572, 30490, 143552, 683101, 3274826, 15796897, 76581875, 372868101, 1822236628, 8934910362, 43939164263, 216651036012
(list)
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OFFSET
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1,3
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REFERENCES
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A. T. Balaban and F. Harary, Chemical graphs V: enumeration and proposed nomenclature of benzenoid cata-condensed polycyclic aromatic hydrocarbons, Tetrahedron 24 (1968), 2505-2506.
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.
F. Harary and R. C. Read, The enumeration of tree-like polyhexes, Proc. Edinb. Math. Soc. (2) 17 (1970), 1-13.
D. A. Klarner, Cell growth problems, Canad. J. Math., 19 (1967), 851-863.
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.
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LINKS
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A. Clarke, Polycubes
D. Gouyou-Beauchamps and P. Leroux, Enumeration of symmetry classes of convex polyominoes on the honeycomb lattice.
M. Keller, Counting polyforms
N. J. A. Sloane, Illustration of initial terms
E. W. Weisstein, Link to a section of The World of Mathematics.
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CROSSREFS
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Cf. A036359, A002216, A005963, A000228, A001998, A018190.
A000228 = (A006535 + A030225)/2.
Cf. A001207, A057973.
Sequence in context: A075214 A070766 A018190 this_sequence A108070 A038147 A082271
Adjacent sequences: A000225 A000226 A000227 this_sequence A000229 A000230 A000231
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KEYWORD
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nonn,nice,hard
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AUTHOR
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njas
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EXTENSIONS
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a(13) from Achim Flammenkamp (achim(AT)uni-bielefeld.de) Feb 15 1999. a(14) from Brendan Owen, Dec 31, 2001
a(15) from Joseph Myers (jsm(AT)polyomino.org.uk), May 05 2002
More terms from Joseph Myers (jsm(AT)polyomino.org.uk), Sep 21 2002
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Note the keyword "hard"!
This indicates that all the terms known to science
are shown - no one knows the next term.
No formula, recurrence or simple algorithm is known for this sequence.
Clicking on the link in the line
Links: N. J. A. Sloane, Illustration of initial terms.
produces the following pictures:
At least this explains what the sequence is about, and suggests
why it is such a hard problem (because of the
requirement that the hexagons must lie in a plane).
Click the single right arrow to go to the next demonstration page,
or the single left arrow to return to the previous page.
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