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A255238
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Triangle T(n, m) of numbers of points of a square lattice covered by a circular disk of radius n (centered at any lattice point taken as origin) with ordinate y = m in the first quadrant.
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4
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1, 2, 1, 3, 2, 1, 4, 3, 3, 1, 5, 4, 4, 3, 1, 6, 5, 5, 5, 4, 1, 7, 6, 6, 6, 5, 4, 1, 8, 7, 7, 7, 6, 5, 4, 1, 9, 8, 8, 8, 7, 7, 6, 4, 1, 10, 9, 9, 9, 9, 8, 7, 6, 5, 1, 11, 10, 10, 10, 10, 9, 9, 8, 7, 5, 1
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OFFSET
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0,2
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COMMENTS
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This entry is motivated by the proposal A255195 by Mats Granvik.
See the MathWorld link on Gauss's circle problem.
The first quadrant of a square lattice (x, y) with x = n >= 0, y = m >= 0, is considered. The number of lattice points covered by a circular disk of radius R = n around the origin having ordinate value y = m are denoted by T(n, m), for n >= 0 and m = 0, 1, ..., n.
The same numbers occur if x and y are interchanged.
One could also consider the row reversed triangle.
The row sums give R(n) = A000603(n), n >= 0.
The alternating row sums give A255239(n), n >= 0.
The total number of square lattice points covered by a circular disk of radius n is A000328(n) = 4*R(n) - (4*n+3).
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LINKS
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FORMULA
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T(n, m) = 1 + floor(sqrt(n^2 - m^2)), 0 <= m <= n.
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EXAMPLE
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The triangle T(n, m) begins:
n\m 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0: 1
1: 2 1
2: 3 2 1
3: 4 3 3 1
4: 5 4 4 3 1
5: 6 5 5 5 4 1
6: 7 6 6 6 5 4 1
7: 8 7 7 7 6 5 4 1
8: 9 8 8 8 7 7 6 4 1
9: 10 9 9 9 9 8 7 6 5 1
10: 11 10 10 10 10 9 9 8 7 5 1
11: 12 11 11 11 11 10 10 9 8 7 5 1
12: 13 12 12 12 12 11 11 10 9 8 7 5 1
13: 14 13 13 13 13 13 12 11 11 10 9 7 6 1
14: 15 14 14 14 14 14 13 13 12 11 10 9 8 6 1
15: 16 15 15 15 15 15 14 14 13 13 12 11 10 8 6 1
...
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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