A367758 Least number of inequivalent cells in a polyomino with n cells.
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, 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, 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
Offset: 1
Keywords
Examples
The X pentomino has 2 inequivalent cells and no pentomino have all cells equivalent, so a(5) = 2.
Links
- Pontus von Brömssen, Illustration of optimal polyominoes for n = 1..13.
- John Mason and Pontus von Brömssen, Proof of formula.
- Index entries for sequences related to polyominoes.
- Index entries for linear recurrences with constant coefficients, signature (1,-1,1,0,0,0,0,1,-1,1,-1).
Formula
a(n) > n/8.
From John Mason and Pontus von Brömssen, Oct 08 2024: (Start)
For n != 1,5, n = 8*k + c, for integers k and c, k >= 0, 0 <= c <= 7:
if c = 0 or 1 then a(n) = k + c + 1;
if c = 2 or 6 then a(n) = 2*k + (c+2)/4;
if c = 3 or 7 then a(n) = 2*k + (c+5)/4;
if c = 4 then a(n) = k + 1;
if c = 5 then a(n) = k + 3.
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.
a(n) = 2*a(n-8) - a(n-16) for n >= 22. (End)
Extensions
a(14)-a(18) from John Mason, Sep 19 2024
More terms from John Mason, Oct 08 2024
Comments