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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.

%I A365964 #36 Mar 06 2025 08:28:13
%S A365964 0,1,4,8,20,33,52,78,108,156,212,264,340,425,528,640,780,925,1084,
%T A365964 1255,1428,1664,1916,2183,2474,2769,3116,3464,3852,4258,4688,5120,
%U A365964 5680,6241,6816,7406,7992,8689,9388,10127,10888,11729,12592,13495,14400,15440,16512
%N 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.
%C A365964 From _Pontus von Brömssen_, Feb 26 2025: (Start)
%C A365964 a(1)-a(40) appear in Brlek, Labelle, and Lacasse (2008).
%C A365964 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.
%C A365964 (End)
%H A365964 Pontus von Brömssen, <a href="/A365964/b365964.txt">Table of n, a(n) for n = 1..67</a>
%H A365964 Srečko Brlek, Gilbert Labelle, and Annie Lacasse, <a href="https://doi.org/10.1016/j.tcs.2008.06.015">Discrete sets with minimal moment of inertia</a>, Theoretical Computer Science 406 (2008), 31-42. See Tables 1-2 and Figure 8.
%H A365964 Pontus von Brömssen, <a href="/A365964/a365964_1.svg">Illustration of the optimal polyominoes for 1 <= n <= 67, with their centers of mass marked with a dot</a>.
%H A365964 Pontus von Brömssen, <a href="https://oeis.org/plot2a?name1=A365964&amp;name2=A000578&amp;tform1=untransformed&amp;tform2=untransformed&amp;shift=0&amp;radiop1=ratio&amp;drawpoints=true">Plot of a(n)/n^3 vs n</a>, using Plot2.
%H A365964 <a href="/index/Mo#moment_of_inertia">Index entries for sequences related to moment of inertia</a>.
%H A365964 <a href="/index/Pol#polyominoes">Index entries for sequences related to polyominoes</a>.
%F A365964 a(n) ~ n^3/(2*Pi).
%e A365964 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.
%e A365964       +---+                   +---+---+
%e A365964       |   |                   |   |   |
%e A365964   +---+---+---+           +---+---+---+
%e A365964   |   |   |   |           |   |   |   |
%e A365964   +---+---+---+---+       +---+---+---+---+
%e A365964   |   |   |   |   |       |   |   |   |   |
%e A365964   +---+---+---+---+       +---+---+---+---+
%e A365964   |   |   |   |               |   |   |
%e A365964   +---+---+---+               +---+---+
%e A365964 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
%Y A365964 Row minima of A365963.
%Y A365964 Cf. A000578.
%K A365964 nonn
%O A365964 1,3
%A A365964 _Pontus von Brömssen_, Sep 23 2023
%E A365964 a(14)-a(16) from _Pontus von Brömssen_, Apr 03 2024
%E A365964 More terms from _Pontus von Brömssen_, Feb 26 2025