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A255195
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Triangle describing the shape of one eighth of the Gauss circle problem.
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4
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1, 2, 0, 2, 1, 0, 2, 1, 1, 0, 2, 1, 2, 0, 0, 2, 1, 1, 2, 0, 0, 2, 1, 1, 2, 1, 0, 0, 2, 1, 1, 2, 2, 0, 0, 0, 2, 1, 1, 2, 1, 2, 0, 0, 0, 2, 1, 1, 1, 2, 2, 1, 0, 0, 0, 2, 1, 1, 1, 2, 1, 2, 1, 0, 0, 0, 2, 1, 1, 1, 2, 1, 2, 2, 0, 0, 0, 0, 2, 1, 1, 1, 2, 1, 2, 2, 1, 0, 0, 0, 0
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OFFSET
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1,2
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COMMENTS
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The sum of terms of row n is n.
Total of partial sums in reverse (from right to left) equals one eighth of the Gauss circle problem. Whenever there is the number 2 the border of the circle makes a jump upwards. Predicting where the 2's are would say something about the Gauss circle problem. The number of 2's equals the number of 0's in the same row, and is counted by A194920(n-1).
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LINKS
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FORMULA
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A000603(n) = 2*(Sum_{k=1..n} Sum_{k=1..k} T(n,n-k+1))-ceiling((n-1)/sqrt(2)) for n>1.
A247588(n-1) = (Sum_{k=1..n} Sum_{k=1..k} (T(n,k) - T(n,n-k+1))/2).
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EXAMPLE
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1,
2, 0,
2, 1, 0,
2, 1, 1, 0,
2, 1, 2, 0, 0,
2, 1, 1, 2, 0, 0,
2, 1, 1, 2, 1, 0, 0,
2, 1, 1, 2, 2, 0, 0, 0,
2, 1, 1, 2, 1, 2, 0, 0, 0,
2, 1, 1, 1, 2, 2, 1, 0, 0, 0,
2, 1, 1, 1, 2, 1, 2, 1, 0, 0, 0,
2, 1, 1, 1, 2, 1, 2, 2, 0, 0, 0, 0,
2, 1, 1, 1, 2, 1, 2, 2, 1, 0, 0, 0, 0
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MATHEMATICA
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Flatten[Table[Sum[Table[If[And[If[n^2 + k^2 <= r^2, If[n >= k, 1, 0], 0] == 1, If[(n + 1)^2 + (k + 1)^2 <= r^2, If[n >= k, 1, 0], 0]== 0], 1, 0], {k, 0, r}], {n, 0, r}], {r, 0, 12}]]
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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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