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A249870
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Rational parts of the Q(sqrt(3)) integers giving the square of the radii for lattice point circles for the Archimedean tiling (3, 4, 6, 4).
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5
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0, 1, 2, 3, 2, 4, 4, 4, 5, 6, 8, 8, 7, 8, 10, 10, 10, 13, 14, 11, 12, 13, 15, 14, 16, 16, 17, 16, 19, 20, 22, 19, 20, 20, 24, 23, 21, 25, 22, 23, 28, 26, 26, 28, 31, 28, 32, 28, 28, 30, 32, 34, 35, 32, 33, 38, 34, 36, 38, 37, 40, 37, 38, 43, 40, 44, 40, 46
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OFFSET
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0,3
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COMMENTS
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The irrational parts are given in A249871.
The points of the lattice of the Archimedean tiling (3, 4, 6, 4) lie on certain circles around any point. The length of the side of the regular 6-gon is taken as 1 (in some length unit).
The squares of the radii R2(n) of these circles are integers in the real quadratic number field Q(sqrt(3)), hence R2(n) = a(n) + A249871(n)*sqrt(3). The R2 sequence is sorted in increasing order.
For details see the notes given in a link.
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LINKS
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EXAMPLE
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The pairs [a(n), A249871(n)] for the squares of the radii R2(n) begin:
[0, 0], [1, 0], [2, 0], [3, 0], [2, 1], [4, 0], [4, 1], [4, 2], [5, 2], [6, 3], [8, 2], [8, 3], [7, 4], [8, 4], [10, 3]] ...
The corresponding radii R(n) are (Maple 10 digits, if not an integer):
0, 1, 1.414213562, 1.732050808, 1.931851653, 2, 2.394170171, 2.732050808, 2.909312911, 3.346065215, 3.385867927, 3.632650881, 3.732050808, 3.863703305, 3.898224265 ...
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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