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A255195
Triangle describing the shape of one eighth of the Gauss circle problem.
4
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
OFFSET
1,2
COMMENTS
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).
FORMULA
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).
EXAMPLE
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
MATHEMATICA
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}]]
CROSSREFS
Sequence in context: A035394 A321101 A067167 * A194317 A096810 A190436
KEYWORD
nonn,tabl
AUTHOR
Mats Granvik, Feb 16 2015
STATUS
approved