OFFSET
1,3
COMMENTS
Two cells in a polyomino are equivalent if there is a symmetry of the polyomino that takes one of the cells to the other.
Equivalently, a(n) is the least number of pointed polyominoes (A126202) corresponding to a given polyomino with n cells.
LINKS
Pontus von Brömssen, Illustration of optimal polyominoes for n = 1..13.
John Mason and Pontus von Brömssen, Proof of formula.
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)
EXAMPLE
The X pentomino has 2 inequivalent cells and no pentomino have all cells equivalent, so a(5) = 2.
CROSSREFS
KEYWORD
nonn
AUTHOR
Pontus von Brömssen, Nov 29 2023
EXTENSIONS
a(14)-a(18) from John Mason, Sep 19 2024
More terms from John Mason, Oct 08 2024
STATUS
approved