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A348469
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Maximal number of squares that can be formed from the grid points in a qualifying circular region of the plane that contains exactly n points of a square grid.
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2
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0, 0, 0, 0, 1, 1, 2, 3, 4, 6, 7, 8, 11, 13, 15, 17, 20, 22, 25, 28, 32, 37, 40, 43, 47, 51, 56, 60, 65, 70, 75, 81, 88, 92, 97, 103, 109, 117, 123, 130, 137, 144, 151, 158, 166, 175, 182, 189, 198, 207, 216, 226, 237, 245, 254, 263, 272, 282, 293, 303, 314
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OFFSET
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0,7
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COMMENTS
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A circular region qualifies if (1) 3 (or more) grid points are incident on its circumference, or (2) it is an adjustment of a circular region, D, as defined in (1), so as to exclude only one or only consecutive grid points on the circumference of D. (Any such points on the circumference of D can be excluded by perturbing the center and radius of D by compatible but arbitrarily small amounts.)
The sequence definition is designed to help investigate the extent to which terms of A051602 can be equalled using only circular regions, while facilitating quicker calculation of terms. At the time of first submission, it is not clear to the authors that the qualification on the circular regions excludes any otherwise permissible configuration of points. In the absence of this knowledge, the qualification allows for the desired quicker calculation.
See A051602 for more information, references and links related to the general problem.
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LINKS
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FORMULA
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EXAMPLE
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For the following examples, we refer to Hugo Pfoertner's pictorial catalog of circles passing through 3 or more grid points (see links section). Each illustration in the catalog is headed by the relevant terms of the sequences that give the squared radii of the circles, e.g. "A192493(5) = 25, A192494(5) = 16". The last line underneath each illustration gives the number of grid points in the circular region, e.g. "4+3=7" indicates 7 grid points total, of which 3 are on the circumference.
For n = 10, in the Pfoertner catalog we see the only circular region with 10 points corresponds to A192493(8). From the points in the illustration for A192493(8), 7 squares can be formed. This matches A051602(10) = 7, the maximal number of squares that can be formed from 10 points, so a(10) = 7.
For n = 20, in the Pfoertner catalog the only circular region with 20 points corresponds to A192493(29). From the points in the illustration for A192493(29), 31 squares can be formed. The circular region corresponding to A192493(24) has 21 points. From the points in the illustration for A192493(24) with any circumferential point excluded to leave 20 points, 32 squares can be formed. From a comprehensive search not detailed here, we ascertain that 32 is the most squares that can be formed from a 20 point configuration defined in the specified manner, so a(20) = 32.
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PROG
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(C++) See Links section. The program's first option generates terms for n <= 100. There are more general options and features, but none designed to generate a larger number of terms automatically.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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