A365964 a(n) is n times the minimum moment of inertia of an n-celled polyomino about an axis through the center of mass perpendicular to the plane of the polyomino, with a unit point mass in the center of each of the cells.

0, 1, 4, 8, 20, 33, 52, 78, 108, 156, 212, 264, 340, 425, 528, 640, 780, 925, 1084, 1255, 1428, 1664, 1916, 2183, 2474, 2769, 3116, 3464, 3852, 4258, 4688, 5120, 5680, 6241, 6816, 7406, 7992, 8689, 9388, 10127, 10888, 11729, 12592, 13495, 14400, 15440, 16512

Offset

1,3

Comments

From Pontus von Brömssen, Feb 26 2025: (Start)

a(1)-a(40) appear in Brlek, Labelle, and Lacasse (2008).

For n = 5, 11, 16, 17, 33, there are two (free) polyominoes with the minimum moment of inertia a(n)/n. For n <= 67, there are never more than two. See linked illustration.

(End)

Links

Pontus von Brömssen, Table of n, a(n) for n = 1..67

Srečko Brlek, Gilbert Labelle, and Annie Lacasse, Discrete sets with minimal moment of inertia, Theoretical Computer Science 406 (2008), 31-42. See Tables 1-2 and Figure 8.

Pontus von Brömssen, Illustration of the optimal polyominoes for 1 <= n <= 67, with their centers of mass marked with a dot.

Pontus von Brömssen, Plot of a(n)/n^3 vs n, using Plot2.

Index entries for sequences related to moment of inertia.

Index entries for sequences related to polyominoes.

Formula

a(n) ~ n^3/(2*Pi).

Example

For some n, there are more than one polyomino that have the minimum possible moment of inertia. For n = 5, for example, both the P-pentomino and the X-pentomino have the minimum possible moment of inertia a(5)/5 = 4; and for n = 11, the two undecominoes below both have the minimum possible moment of inertia a(11)/11 = 212/11.
      +---+                   +---+---+
      |   |                   |   |   |
  +---+---+---+           +---+---+---+
  |   |   |   |           |   |   |   |
  +---+---+---+---+       +---+---+---+---+
  |   |   |   |   |       |   |   |   |   |
  +---+---+---+---+       +---+---+---+---+
  |   |   |   |               |   |   |
  +---+---+---+               +---+---+
Also for n = 16 there are two polyominoes with the minimum moment of inertia a(16)/16 = 40: the 4 X 4 square and the 5 X 4 square with the corner cells removed. - Pontus von Brömssen, Apr 03 2024

Crossrefs

Row minima of A365963.

Cf. A000578.

Sequence in context: A047185 A349546 A034733 * A152233 A301896 A053303

Adjacent sequences: A365961 A365962 A365963 * A365965 A365966 A365967

Keyword

nonn,changed

Author

Pontus von Brömssen, Sep 23 2023

Extensions

a(14)-a(16) from Pontus von Brömssen, Apr 03 2024

More terms from Pontus von Brömssen, Feb 26 2025

Status

approved