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

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

Author

Gordon Hamilton, Mar 09 2015

Keywords

• nonn
• more

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