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 1-7 of 7 results.

A367759 Number of free n-celled polyominoes with the least number (A367758(n)) of inequivalent cells.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 2, 3, 2, 3, 4, 2, 5, 10, 10, 4, 2, 33
Offset: 1

Views

Author

Pontus von Brömssen, Nov 29 2023

Keywords

Links

Crossrefs

Cf. A367758.

Extensions

a(14)-a(18) from John Mason, Sep 19 2024

A255890 Polyomino Family Planners: a(n) is the least number of children of a polyomino of size n.

Original entry on oeis.org

1, 1, 2, 3, 1, 2, 3, 3, 3, 2, 3, 4, 2, 2, 4, 4, 2, 3, 4, 4, 3, 3, 5, 4, 2, 3, 5, 5, 3, 3, 5, 6, 3, 3, 5, 6, 3
Offset: 0

Views

Author

Gordon Hamilton, Mar 09 2015

Keywords

Comments

For n = (2k+1)^2 + (2k)^2, a(n) = k+1 and a(m) > k+1 for m > n.
This is a beautiful exploration of symmetry for the elementary classroom.
A "child" is any polyomino formed by adjoining a cell at any edge. - N. J. A. Sloane, Mar 10 2015
Optimal polyominoes have at least fourfold symmetry. - Charlie Neder, Mar 03 2019

Examples

			a(7) = 3 because this polyomino has only three children:
   xx                        xxx      xx       xx
   xxx      has children     xxx      xxxx     xxx
    xx                        xx       xx       xxx
a(8) = 3 because of this polyomino:
                             xxxx
                             xxxx
a(9) = 2 because of this polyomino:
                             xxx
                             xxx
                             xxx
a(10) = 3 because of this polyomino (not the 2*5 rectangle):
                             xx
                             xxx
                              xxx
                               xx
a(11) = 4 because of this polyomino:
                             xxx
                            xxxxx
                             xxx
a(12) = 2 because of this polyomino:
                             xx
                            xxxx
                            xxxx
                             xx
a(13) = 2 because of the following polyomino. This will be the last time 2 will be encountered in the sequence (see comments above):
                              x
                             xxx
                            xxxxx
                             xxx
                              x
a(14) = 4 because of this polyomino:
                             xxx
                            xxxx
                            xxxx
                            xxx
a(15) = 4 because of this polyomino:
                            xx
                            xxxx
                             xxx
                             xxxx
                               xx

Crossrefs

Cf. A000105, A255894, A367758.
Row minima of A367443 (for n>=1).

Formula

From Charlie Neder, Mar 03 2019: (Start)
a(4k) >= b, where b is the least integer such that b(2b-1) >= k.
a(4k+1) = c, where c is the least integer such that (c-1)(2c-1) >= k. (End)

Extensions

a(16)-a(36) from Charlie Neder, Mar 03 2019

A369363 Least number of inequivalent cells in a (planar) polyhex with n cells.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 3, 2, 3, 4, 2, 3, 4, 3, 4
Offset: 1

Views

Author

Pontus von Brömssen, Jan 22 2024

Keywords

Comments

Equivalently, a(n) is the least number of free rooted (or pointed) polyhexes (A369362) corresponding to a given polyhex with n cells.

Links

Crossrefs

Cf. A000228, A367758, A369362, A369364, A369366.

Extensions

a(13)-a(16) from John Mason, Sep 24 2024

A369366 Least number of inequivalent cells in a polyiamond with n cells.

Original entry on oeis.org

1, 1, 2, 2, 3, 1, 3, 3, 3, 3, 6, 2, 4, 4, 4, 4, 9, 2
Offset: 1

Views

Author

Pontus von Brömssen, Jan 22 2024

Keywords

Comments

Equivalently, a(n) is the least number of free rooted (or pointed) polyiamonds (A369365) corresponding to a given polyiamond with n cells.

Links

Crossrefs

Cf. A000577, A367758, A369363, A369365, A369367.

Extensions

a(18) from John Mason, Sep 20 2024

A126202 Number of pointed polyominoes with n cells.

Original entry on oeis.org

1, 1, 1, 4, 12, 46, 172, 689, 2763, 11241, 45706, 186346, 759375, 3095191, 12610825, 51370336, 209194923, 851712069, 3466881944
Offset: 0

Views

Author

David Applegate and N. J. A. Sloane, Mar 09 2007

Keywords

Comments

Number of pairs (P,C), where P is a polyomino with n cells (cf. A000105) and C runs through the orbit representatives of cells in P under the automorphism group of P.

Examples

			a(4) = 12: polyomino and number of orbits on points:
XXXX..XXX..XXX..XX...XX
......X.....X....XX..XX
(2)...(4)..(3)..(2)..(1): total 12

Crossrefs

Cf. A000105, A367758, A367759.

A376798 Least number of inequivalent cells in a polycube with n cells.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 2, 1, 3, 2, 3, 2, 3
Offset: 1

Views

Author

John Mason, Oct 04 2024

Keywords

Comments

a(27) <= 4, a(64) <= 4.

Links

Crossrefs

Cf. A367758, A376797, A376799.

Formula

a(n) <= A367758(n).
From Pontus von Brömssen, Oct 13 2024: (Start)
n/48 < a(n) <= n/48 + c for some constant c. c = 11 + 19/48 works for n >= 78, but that is not best possible. See link "Inequivalent cells in polycubes".
Conjecture: The sequence (a(n)-n/48) is eventually periodic with period 48.
(End)

A373635 Number of free n-celled polyominoes to which two inequivalent cells can be adjoined such that the two resulting free (n+1)-celled polyominoes are identical.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 0, 0, 4, 6, 2, 0, 17, 29
Offset: 1

Views

Author

Pontus von Brömssen, Jun 13 2024

Keywords

Comments

Two cells that can be adjoined to an n-celled polyomino are equivalent if there is an isomorphism between the two resulting (n+1)-celled polyominoes that maps the adjoined cell of the first to the adjoined cell of the second.

Examples

			The smallest example is the P pentomino shown below. The two free hexominoes obtained by adjoining one of the two cells marked "*" are identical, but there is no isomorphism between them that also maps the marked cell of the first to the marked cell of the second.
    _ _
  *|   |
   |  _|*
   |_|

Links

Crossrefs

Cf. A255890, A367758, A373636, A373637.
Showing 1-7 of 7 results.

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

Content is available under The OEIS End-User License Agreement.