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Irregular table read by rows: T(n,k) is the number of k-gons, k>=2, among all distinct circles that can be constructed from an n x n square grid of points using only a compass.

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`%I #12 Jan 04 2024 14:45:51
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`%S 0,16,30,0,412,341,60,20,4,0,3464,3534,928,212,48,12,0,16936,19861,
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`%T 5252,1056,88,52,8,0,63712,77394,20480,4820,612,108,20,12,4,202904,
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`%U 244013,71244,14968,1852,472,80,32,4
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`%N Irregular table read by rows: T(n,k) is the number of k-gons, k>=2, among all distinct circles that can be constructed from an n x n square grid of points using only a compass.
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`%C A circle is constructed for every pair of the n x n points, the first point defines the circle's center while the second the radius distance. The number of distinct circles constructed from the n x n points is A359931(n).
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`%C See A359932 and A359933 for images of the circles.
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`%C The first occurrence of a 2-gon is when n = 7. Assuming the grid points are separated by 1 unit, in the first quadrant this region has endpoints (6,7) and (7,6) - an equivalent region is in each of the three other quadrants. Its arcs are from two circles, one with center at (2,2) going through point (-2,-3) while the other has center (3,3) going through point (0,-1). See the attached image.
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`%H Scott R. Shannon, <a href="/A359935/a359935.jpg">Close-up of the circles for n = 7</a>. The 2-gon is the thin region shown in white in the upper-right corner.
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`%F Sum of row n = A359933(n).
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`%e The table begins:
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`%e 0, 16, 30;
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`%e 0, 412, 341, 60, 20, 4;
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`%e 0, 3464, 3534, 928, 212, 48, 12;
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`%e 0, 16936, 19861, 5252, 1056, 88, 52, 8;
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`%e 0, 63712, 77394, 20480, 4820, 612, 108, 20, 12;
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`%e 4, 202904, 244013, 71244, 14968, 1852, 472, 80, 32, 4;
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`%e .
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`%e .
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`%Y Cf. A359932 (vertices), A359933 (regions), A359934 (edges), A359931 (distinct circles), A359862, A359258.
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`%K nonn,tabf,more
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`%O 2,2
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`%A _Scott R. Shannon_, Jan 21 2023
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