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
Keywords
Examples
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.
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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
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).
Extensions
a(14)-a(16) from Pontus von Brömssen, Apr 03 2024
More terms from Pontus von Brömssen, Feb 26 2025
Comments