cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A057730 Number of polyominoes (A000105) with perimeter 2n.

Original entry on oeis.org

0, 1, 1, 3, 6, 25, 86, 416, 1988, 10640, 57987, 328956, 1900321, 11204350, 67042778, 406780346
Offset: 1

Views

Author

N. J. A. Sloane, Oct 29 2000

Keywords

Comments

Does this include polyominoes with holes? - Franklin T. Adams-Watters, Sep 12 2006. Answer from R. J. Mathar: Yes! See the illustrations in the links (e.g. perimeter 16, area 7, No 81 or perimeter 16, area 8, No 174).
All lines (sides of cells which are not common to a pair of cells) contribute to the perimeter, including the interior sides of cavities and holes. - R. J. Mathar, Feb 19 2021

Links

Crossrefs

Cf. A000105, A002931, A057753, A266549 (same, but holes not allowed), column sums of A342243, A131487 (polyominoes by total number of edges).

Extensions

Additional comments from Barry Cipra, Jun 08 2004
Link updated by William Rex Marshall, Dec 16 2009
a(9)-a(10) added by Luca Petrone, Jan 08 2016
a(1)-a(9) confirmed by Bert Dobbelaere, Oct 19 2018
a(10)-a(12) corrected and extended by John Mason, Jul 26 2021
a(13)-a(16) added by John Mason, Sep 08 2022

A342243 Triangle T(n,p) read by rows: the number of n-celled polyominoes with perimeter 2p, 2 <= p <= 1+n.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 1, 4, 0, 0, 0, 1, 11, 0, 0, 0, 1, 7, 27, 0, 0, 0, 0, 4, 21, 83, 0, 0, 0, 0, 2, 21, 91, 255, 0, 0, 0, 0, 1, 9, 89, 339, 847, 0, 0, 0, 0, 0, 6, 67, 393, 1360, 2829, 0, 0, 0, 0, 0, 1, 45, 325, 1713, 5255, 9734, 0, 0, 0, 0, 0, 1, 23, 275
Offset: 1

Views

Author

R. J. Mathar, Mar 07 2021

Keywords

Examples

			The triangle has rows n=1,2,3,... and columns p=2,3,4,5,...:
  1;
  0, 1;
  0, 0, 2;
  0, 0, 1, 4;
  0, 0, 0, 1, 11;
  0, 0, 0, 1,  7, 27;
  0, 0, 0, 0,  4, 21, 83;
  0, 0, 0, 0,  2, 21, 91, 255;
  0, 0, 0, 0,  1,  9, 89, 339,  847;
  0, 0, 0, 0,  0,  6, 67, 393, 1360, 2829;
  0, 0, 0, 0,  0,  1, 45, 325, 1713, 5255, 9734;
  ...

Links

Crossrefs

Cf. A000105 (row sums), A057730 (column sums), A131482 (diagonal), A131487 (skew antidiagonal sums), A027709 (number of leading zeros per row), A100092 (first nonzero in each row).

Formula

A131487(e) = Sum_{e=2*n+p} T(n,p).

A131488 a(n) is the number of polyhexes with n edges, including inner edges.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 1, 1, 5, 0, 1, 3, 6, 12, 3, 4, 14, 26, 39, 10, 25, 70, 116, 139, 67, 152, 347, 514, 567, 414, 884, 1744, 2408, 2561, 2498, 4967
Offset: 1

Views

Author

Tanya Khovanova, Jul 28 2007

Keywords

Comments

An n-celled polyhex with perimeter p has (6n+p)/2 edges. The maximum number of edges in an n-celled polyhex is 5n+1.
Given Clarke's table T(p,n), a(n) is an antidiagonal sum selecting entries in a (1,3)-leaper's moves. - R. J. Mathar, Feb 23 2021

Examples

			a(31) = T(p=26,A=6) + T(p=20,A=7) = 36+3 = 39. a(34) = T(p=26,A=7) + T(p=20,A=8) = 69+1 = 70. a(35) = 107+9. a(36) = 118+21. a(41) = 411+155+1. a(44) = 1621 +123. a(45) = 1986+420+2. a(46) = 1489+1046+26. - _R. J. Mathar_, Feb 23 2021

Links

Crossrefs

Cf. A000228: Number of hexagonal polyominoes (or planar polyhexes) with n cells. A057779: Hexagonal polyominoes (or polyhexes, A000228) with perimeter 2n. A038142: Number of planar cata-polyhexes with n cells. A131487: analog for square tiling.

Extensions

Extended to a(48). - R. J. Mathar, Feb 23 2021
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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