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A367758 Least number of inequivalent cells in a polyomino with n cells.

%I A367758 #36 Oct 08 2024 18:38:54
%S A367758 1,1,2,1,2,2,3,2,3,3,4,2,4,4,5,3,4,5,6,3,5,6,7,4,5,7,8,4,6,8,9,5,6,9,
%T A367758 10,5,7,10,11,6,7,11,12,6,8,12,13,7,8,13,14,7,9,14,15,8,9,15,16,8,10,
%U A367758 16,17,9,10,17,18,9,11,18,19,10,11,19,20,10,12,20,21,11,12,21,22,11,13,22,23
%N A367758 Least number of inequivalent cells in a polyomino with n cells.
%C A367758 Two cells in a polyomino are equivalent if there is a symmetry of the polyomino that takes one of the cells to the other.
%C A367758 Equivalently, a(n) is the least number of pointed polyominoes (A126202) corresponding to a given polyomino with n cells.
%H A367758 Pontus von Brömssen, <a href="/A367759/a367759.svg">Illustration of optimal polyominoes for n = 1..13</a>.
%H A367758 John Mason and Pontus von Brömssen, <a href="/A367758/a367758.pdf">Proof of formula</a>.
%H A367758 <a href="/index/Pol#polyominoes">Index entries for sequences related to polyominoes</a>.
%H A367758 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (1,-1,1,0,0,0,0,1,-1,1,-1).
%F A367758 a(n) > n/8.
%F A367758 From _John Mason_ and _Pontus von Brömssen_, Oct 08 2024: (Start)
%F A367758 For n != 1,5, n = 8*k + c, for integers k and c, k >= 0, 0 <= c <= 7:
%F A367758   if c = 0 or 1 then a(n) =   k + c + 1;
%F A367758   if c = 2 or 6 then a(n) = 2*k + (c+2)/4;
%F A367758   if c = 3 or 7 then a(n) = 2*k + (c+5)/4;
%F A367758   if c = 4      then a(n) =   k + 1;
%F A367758   if c = 5      then a(n) =   k + 3.
%F A367758 a(n) = a(n-1) - a(n-2) + a(n-3) + a(n-8) - a(n-9) + a(n-10) - a(n-11) for n >= 17.
%F A367758 a(n) = 2*a(n-8) - a(n-16) for n >= 22. (End)
%e A367758 The X pentomino has 2 inequivalent cells and no pentomino have all cells equivalent, so a(5) = 2.
%Y A367758 Cf. A126202, A255890, A367759, A369363, A369366, A376798.
%K A367758 nonn
%O A367758 1,3
%A A367758 _Pontus von Brömssen_, Nov 29 2023
%E A367758 a(14)-a(18) from _John Mason_, Sep 19 2024
%E A367758 More terms from _John Mason_, Oct 08 2024