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A255890
Polyomino Family Planners: a(n) is the least number of children of a polyomino of size n.
5
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
0,3
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
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)
EXAMPLE
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
Row minima of A367443 (for n>=1).
Sequence in context: A305390 A344310 A362068 * A194300 A065365 A096137
KEYWORD
nonn,more
AUTHOR
Gordon Hamilton, Mar 09 2015
EXTENSIONS
a(16)-a(36) from Charlie Neder, Mar 03 2019
STATUS
approved