cp's OEIS Frontend

A geometric interpretation of the sequence is the number of added or subtracted squares along the edge of (not completely within) an origin centered circle in a quadrant of a Cartesian grid as the radius increases. The number of squares increase or decrease when the radius squared changes from being exactly on a corner of a square (r^2 = m^2+n^2) to the open interval between corners given by (m^2+n^2,(m+1)^2+(n+1)^2). The square radii that correspond to corners are given by A001481, so each a(n) corresponds to the radius changing from a point to an element of an open set bounded by adjacent elements of A001481.

a(n) is 0 when the radius squared increases from the open interval less than a perfect square to the perfect square itself (corresponding to a radius that intersects the x and y axes at an integer), see below for example.

a(n) is odd when the square radius changes to or from an integer which is twice a square integer (on a corner on the y= x line), see below for example.

The positions reflect radii which are a unique sum of two and only two distinct nonzero square integers.

The positions are a bit less frequent in occurrence than the positions where the first differences equal 2 because when the radius changes from exactly an integer value k to the open interval (k,k+1), the number of edge squares increase by 2, while in the reverse case, an increase from the open interval (k,k+1) to exactly k+1, the number of edge squares stays the same rather than decreasing by 2 as occurs in cases when the radii are a sum of two and only two distinct nonzero square integers. This is in contrast to positions where the first difference of A234300 equals 1 which are exactly balanced by positions which equal -1.

The positions reflect radii which are a unique sum of two distinct square integers where order doesn't matter.

The positions are more frequent in occurrence than the positions where the first differences equal -2 because when the radius changes from exactly an integer value k to the open interval (k,k+1), the number of edge squares increases by 2, while in the reverse case, an increase from the open interval (k,k+1) to exactly k+1, the number of edge squares stays the same. This is in contrast to positions where the first difference equals 1 which are exactly balanced by positions which equal -1 .

The positions reflect square radii which are uniquely twice a square integer, that is, the completion of only one square on the y = x line.

The positions reflect square radii which are uniquely twice a square integer, that is, the inclusion of only one square on the y = x line.

A(n) alternates between the numbers for circles which intersect points on the A2 lattice and the numbers for circles which pass in between the points on a lattice.

Draw a circle on a 2D square grid centered at the origin with a radius squared equal to the norm of the Gaussian integers A001481(n). See the images in the links. This sequence gives the number of unit cells intersected by the circumference of the circle. Equivalently this is the number of intersections of the circumference with the x and y integer grid lines.