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A348233 List of distinct squared distances from all points of Don Wilkinson's 123-circle packing to a fixed type (a) point. 0

%I #17 Dec 16 2021 20:12:17

%S 0,9,16,36,52,64,73,81,100,144,145,160,180,208,225,256,265,288,289,

%T 292,324,337,340,388,400,436,441,468,481,505,544,576,580,585,592,612,

%U 640,657,697,720,724,729,784,793,801,820,832,900,916,928,964,976,985

%N List of distinct squared distances from all points of Don Wilkinson's 123-circle packing to a fixed type (a) point.

%C 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.

%C 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.

%C The present sequence lists the exponents in the theta series with respect to a type (a) point.

%C 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.

%H N. J. A. Sloane, <a href="/A348227/a348227_3.pdf">Graph formed by centers of Wilkinson's 123-circle packing</a> (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.

%e The point we start from is of course at distance 0 from itself, so a(1) = 0.

%e The closest points to a type (a) point are the two type (b) points at distance 3, so a(2) = 3^2 = 9.

%e The next-closest are the two type (c) points at distance 4, so a(3) = 4^2 = 16.

%e And so on.

%Y Cf. A348227-A348239.

%K nonn

%O 1,2

%A _N. J. A. Sloane_, Oct 08 2021

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