A006534 Number of one-sided triangular polyominoes (n-iamonds) with n cells; turning over not allowed, holes are allowed. (Formerly M3287)

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

1,4

Comments

The figures are formed by connecting n regular triangles by edges.

"Turning over not allowed" means that axial symmetric polyiamonds are counted separately, thus a(4) = 4 and a(5) = 6 while A000577(4) = 3 and A000577(5) = 4, cf. examples. - M. F. Hasler, Nov 12 2017

References

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.

Links

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.

Example

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)

Crossrefs

Cf. A000577 (same with "turning over allowed"), A030223, A030224, A001420.

Sequence in context: A343677 A222379 A202618 * A064035 A010364 A110391

Adjacent sequences: A006531 A006532 A006533 * A006535 A006536 A006537

Keyword

nonn,hard,nice

Author

N. J. A. Sloane

Extensions

Corrected and extended by David W. Wilson

a(19) from Achim Flammenkamp, Feb 15 1999

a(20) to a(28) from Joseph Myers, Sep 24 2002

Edited by M. F. Hasler, Nov 12 2017

More terms from John Mason, Oct 28 2023

Status

approved