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A342954
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Number of interior regions formed by drawing n unit circles whose centers (s,t) are the partitions of k into 2 parts, where 0 < s <= t and s + t = k, ordered by increasing values of s and where k = 2,3,... .
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0
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1, 3, 6, 11, 14, 21, 24, 31, 36, 39, 46, 53, 56, 63, 70, 75, 78, 85, 92, 99, 102, 109, 116, 123, 128, 131, 138, 145, 152, 159, 162, 169, 176, 183, 190, 195, 198, 205, 212, 219, 226, 233, 236, 243, 250, 257, 264, 271, 276, 279, 286, 293, 300, 307, 314, 321, 324, 331, 338
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OFFSET
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1,2
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LINKS
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FORMULA
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a(1) = 1; a(n) = Sum_{i=1..n-1} (7 - 4*(floor(sqrt(i)) - floor(sqrt(i - 1)) + (floor(sqrt(i + 1) + 1/2) - floor(sqrt(i) + 1/2)) - 2*(floor(sqrt(i + 1)) - floor(sqrt(i)))) for n > 1.
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EXAMPLE
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---------------------------------------------------------------------------
partitions of k into two parts
---------------------------------------------------------------------------
k 2 3 4 5 6 7 8 9
[1,7] [1,8]
[1,5] [1,6] [2,6] [2,7]
[1,3] [1,4] [2,4] [2,5] [3,5] [3,6]
[1,1] [1,2] [2,2] [2,3] [3,3] [3,4] [4,4] [4,5]
----------------------------------------------------------------------------
n circle centers number of regions formed
----------------------------------------------------------------------------
1 (1,1) 1
2 (1,1), (1,2) 3
3 (1,1), (1,2), (1,3) 6
4 (1,1), (1,2), (1,3), (2,2) 11
5 (1,1), (1,2), (1,3), (2,2), (1,4) 14
6 (1,1), (1,2), (1,3), (2,2), (1,4), (2,3) 21
...
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MATHEMATICA
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Join[{1}, Table[Sum[7 - 4 (Floor[Sqrt[i]] - Floor[Sqrt[i - 1]] + Floor[Sqrt[i + 1] + 1/2] - Floor[Sqrt[i] + 1/2]) - 2 (Floor[Sqrt[i + 1]] - Floor[Sqrt[i]]), {i, n - 1}], {n, 2, 80}]]
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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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