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A255195 Triangle describing the shape of one eighth of the Gauss circle problem.

%I A255195 #34 Feb 16 2025 08:33:25
%S A255195 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,
%T A255195 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,
%U A255195 1,1,2,1,2,2,0,0,0,0,2,1,1,1,2,1,2,2,1,0,0,0,0
%N A255195 Triangle describing the shape of one eighth of the Gauss circle problem.
%C A255195 The sum of terms of row n is n.
%C A255195 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).
%H A255195 E. W. Weisstein, <a href="https://mathworld.wolfram.com/GausssCircleProblem.html">Gauss's Circle Problem</a>
%F A255195 A000603(n) = 2*(Sum_{k=1..n} Sum_{k=1..k} T(n,n-k+1))-ceiling((n-1)/sqrt(2)) for n>1.
%F A255195 A247588(n-1) = (Sum_{k=1..n} Sum_{k=1..k} (T(n,k) - T(n,n-k+1))/2).
%e A255195 1,
%e A255195 2, 0,
%e A255195 2, 1, 0,
%e A255195 2, 1, 1, 0,
%e A255195 2, 1, 2, 0, 0,
%e A255195 2, 1, 1, 2, 0, 0,
%e A255195 2, 1, 1, 2, 1, 0, 0,
%e A255195 2, 1, 1, 2, 2, 0, 0, 0,
%e A255195 2, 1, 1, 2, 1, 2, 0, 0, 0,
%e A255195 2, 1, 1, 1, 2, 2, 1, 0, 0, 0,
%e A255195 2, 1, 1, 1, 2, 1, 2, 1, 0, 0, 0,
%e A255195 2, 1, 1, 1, 2, 1, 2, 2, 0, 0, 0, 0,
%e A255195 2, 1, 1, 1, 2, 1, 2, 2, 1, 0, 0, 0, 0
%t A255195 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}]]
%Y A255195 Cf. A000603, A194920.
%K A255195 nonn,tabl
%O A255195 1,2
%A A255195 _Mats Granvik_, Feb 16 2015