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A348233
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List of distinct squared distances from all points of Don Wilkinson's 123-circle packing to a fixed type (a) point.
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0
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0, 9, 16, 36, 52, 64, 73, 81, 100, 144, 145, 160, 180, 208, 225, 256, 265, 288, 289, 292, 324, 337, 340, 388, 400, 436, 441, 468, 481, 505, 544, 576, 580, 585, 592, 612, 640, 657, 697, 720, 724, 729, 784, 793, 801, 820, 832, 900, 916, 928, 964, 976, 985
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OFFSET
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1,2
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COMMENTS
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Wilkinson's 123-circle packing (that is my name for it) is a packing of non-overlapping circles in the plane, and can be seen in the links in A348227. There are three sizes of circles: (a) radius 1, (b) radius 2, and (c) radius 3. See A348227 for further information.
A convenient set of coordinates for the centers are: (a) radius 1: the points (8*i, 6*j), (b) radius 2: the points (8*i, 6*j+3), and (c) radius 3: the points (8*i+4, 6*j), where i and j take all integer values.
The present sequence lists the exponents in the theta series with respect to a type (a) point.
This theta series begins 1 + 2*q^9 + 2*q^16 + 2*q^36 + 4*q^52 + 2*q^64 + 4*q^73 + 2*q^81 + 4*q^100 + 4*q^144 + 4*q^145 + 4*q^160 + 4*q^180 + ... but the terms are too sparse for an OEIS entry.
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LINKS
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N. J. A. Sloane, Graph formed by centers of Wilkinson's 123-circle packing (type (a), black: center of circle of radius 1; type (b), green: center of circle of radius 2; type (c), red: center of circle of radius radius 3). This figure should be rotated counterclockwise by 90 degrees in order to match the other figures in A348227.
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EXAMPLE
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The point we start from is of course at distance 0 from itself, so a(1) = 0.
The closest points to a type (a) point are the two type (b) points at distance 3, so a(2) = 3^2 = 9.
The next-closest are the two type (c) points at distance 4, so a(3) = 4^2 = 16.
And so on.
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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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