cp's OEIS Frontend

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

%I A255890 #30 Apr 26 2024 20:33:39
%S A255890 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,
%T A255890 5,6,3
%N A255890 Polyomino Family Planners: a(n) is the least number of children of a polyomino of size n.
%C A255890 For n = (2k+1)^2 + (2k)^2, a(n) = k+1 and a(m) > k+1 for m > n.
%C A255890 This is a beautiful exploration of symmetry for the elementary classroom.
%C A255890 A "child" is any polyomino formed by adjoining a cell at any edge. - _N. J. A. Sloane_, Mar 10 2015
%C A255890 Optimal polyominoes have at least fourfold symmetry. - _Charlie Neder_, Mar 03 2019
%F A255890 From _Charlie Neder_, Mar 03 2019: (Start)
%F A255890 a(4k) >= b, where b is the least integer such that b(2b-1) >= k.
%F A255890 a(4k+1) = c, where c is the least integer such that (c-1)(2c-1) >= k. (End)
%e A255890 a(7) = 3 because this polyomino has only three children:
%e A255890    xx                        xxx      xx       xx
%e A255890    xxx      has children     xxx      xxxx     xxx
%e A255890     xx                        xx       xx       xxx
%e A255890 a(8) = 3 because of this polyomino:
%e A255890                              xxxx
%e A255890                              xxxx
%e A255890 a(9) = 2 because of this polyomino:
%e A255890                              xxx
%e A255890                              xxx
%e A255890                              xxx
%e A255890 a(10) = 3 because of this polyomino (not the 2*5 rectangle):
%e A255890                              xx
%e A255890                              xxx
%e A255890                               xxx
%e A255890                                xx
%e A255890 a(11) = 4 because of this polyomino:
%e A255890                              xxx
%e A255890                             xxxxx
%e A255890                              xxx
%e A255890 a(12) = 2 because of this polyomino:
%e A255890                              xx
%e A255890                             xxxx
%e A255890                             xxxx
%e A255890                              xx
%e A255890 a(13) = 2 because of the following polyomino. This will be the last time 2 will be encountered in the sequence (see comments above):
%e A255890                               x
%e A255890                              xxx
%e A255890                             xxxxx
%e A255890                              xxx
%e A255890                               x
%e A255890 a(14) = 4 because of this polyomino:
%e A255890                              xxx
%e A255890                             xxxx
%e A255890                             xxxx
%e A255890                             xxx
%e A255890 a(15) = 4 because of this polyomino:
%e A255890                             xx
%e A255890                             xxxx
%e A255890                              xxx
%e A255890                              xxxx
%e A255890                                xx
%Y A255890 Cf. A000105, A255894, A367758.
%Y A255890 Row minima of A367443 (for n>=1).
%K A255890 nonn,more
%O A255890 0,3
%A A255890 _Gordon Hamilton_, Mar 09 2015
%E A255890 a(16)-a(36) from _Charlie Neder_, Mar 03 2019