I played around with Gauss circle problem and found that if you take a certain sum in reverse and "in forward" and subtract the resulting sequences you get the OEIS sequence:
starting:
1, 2, 3, 5, 6, 8, 11, 13, 15, 17, 21, 25, 27, 31, 34, 39, 43, 48, ...
My question is: What is the shape of the lattice point set that is counted by this sequence?
Introduction:
Consider the number triangle defined by the Mathematica program:
nn = 12;
(*Mathematica program 1 start*)(*The number triangle above*)TableForm[
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, nn}]]
(*Mathematica program 1 end*)
This an infinite lower triangular matrix starting:
$$\begin{array}{lllllllllllll} 1 & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\ 2 & 0 & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\ 2 & 1 & 0 & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\ 2 & 1 & 1 & 0 & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\ 2 & 1 & 2 & 0 & 0 & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\ 2 & 1 & 1 & 2 & 0 & 0 & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\ 2 & 1 & 1 & 2 & 1 & 0 & 0 & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\ 2 & 1 & 1 & 2 & 2 & 0 & 0 & 0 & \text{} & \text{} & \text{} & \text{} & \text{} \\ 2 & 1 & 1 & 2 & 1 & 2 & 0 & 0 & 0 & \text{} & \text{} & \text{} & \text{} \\ 2 & 1 & 1 & 1 & 2 & 2 & 1 & 0 & 0 & 0 & \text{} & \text{} & \text{} \\ 2 & 1 & 1 & 1 & 2 & 1 & 2 & 1 & 0 & 0 & 0 & \text{} & \text{} \\ 2 & 1 & 1 & 1 & 2 & 1 & 2 & 2 & 0 & 0 & 0 & 0 & \text{} \\ 2 & 1 & 1 & 1 & 2 & 1 & 2 & 2 & 1 & 0 & 0 & 0 & 0 \end{array}$$
It might be of importance that the row sums are equal to the natural numbers.
Then take the total of the partial sums "in forward".
(*Mathematica program 2 start*)
(*How the number triangle above relates to one eight of a circle*)
a2 = Table[
Total[Accumulate[
Reverse[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, nn}]
(*Mathematica program 2 end*)
The output from the program is the sequence:
1, 2, 4, 7, 10, 15, 20, 25, 32, 40, 49, 57, 66,...
which is the same as the number of lattice points within one eight of Gauss circle problem.
Comparison:
(*Mathematica program 3 start*)
(*One eight of a circle the usual way for comparison*)
Table[Sum[
Sum[If[n^2 + k^2 <= r^2, If[n >= k, 1, 0], 0], {k, 0, r}], {n, 0,
r}], {r, 0, nn}]
(*Mathematica program 3 end*)
Output: 1, 2, 4, 7, 10, 15, 20, 25, 32, 40, 49, 57, 66,...
Then take the total of the partial row sums in reverse of the lower triangular number triangle at the beginning of the question.
(*Mathematica program 4 start*)
(*How the number triangle above relates to the number of integer \
sided triangles*)
a4 = Table[
Total[Accumulate[
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, nn}]
(*Mathematica program 4 end*)
Output: 1, 4, 8, 13, 20, 27, 36, 47, 58, 70, 83, 99, 116,...
Then take the difference of the a2 and a4 and divide it by 2:
In other words, this sequence: 1, 4, 8, 13, 20, 27, 36, 47, 58, 70, 83, 99, 116,... minus this sequence: 1, 2, 4, 7, 10, 15, 20, 25, 32, 40, 49, 57, 66,... divided by 2, which is:
1, 2, 3, 5, 6, 8, 11, 13, 15, 17, 21, 25, 27, 31, 34, 39, 43, 48, ...
What is then the geometric, figurative number, meaning of this sequence
as it relates to Gauss circle problem?
I know from the OEIS that it counts integer sided acute triangles, but it should also count some lattice point set within a circle.