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

1,3

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

Paolo Xausa, Table of n, a(n) for n = 1..10000

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)

G.f.: x*(1 + 2*x^2 - x^3 + 2*x^4 - x^5 + 2*x^6 - x^7 + x^8 - x^9 - x^11 + x^12 - x^13 + x^14 - x^15)/(1 - x + x^2 - x^3 - x^8 + x^9 - x^10 + x^11). - Charles R Greathouse IV, May 17 2026

Example

The X pentomino has 2 inequivalent cells and no pentomino have all cells equivalent, so a(5) = 2.

Mathematica

LinearRecurrence[{1, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1}, {1, 1, 2, 1, 2, 2, 3, 2, 3, 3, 4, 2, 4, 4, 5, 3}, 100] (* Paolo Xausa, Mar 03 2026 *)

Crossrefs

Cf. A126202, A255890, A367759, A369363, A369366, A376798.

Sequence in context: A232548 A029363 A304708 * A358370 A319688 A033922

Adjacent sequences: A367755 A367756 A367757 * A367759 A367760 A367761

Keyword

nonn,easy

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