A235141 First differences of A234300.

1, 0, 2, -1, 1, 0, 2, -2, 2, -1, 1, 0, 2, -2, 2, -2, 2, 0, 2, -2, 2, -1, 1, -2, 2, -2, 4, -2, 2, -2, 2, -1, 1, -2, 2, 0, 2, -2, 2, -2, 2, -2, 2, -2, 2, 0, 2, -3, 3, -2, 2, -2, 2, -2, 2, -2, 2, 0, 2, -4, 4, -2, 2, -1, 1, -2, 2, -2, 2, -2, 2, 0, 2, -2, 2, -4, 4, -2, 2, -2, 2

Offset: 1

Author

Rajan Murthy, Jan 03 2014

Keywords

• sign

Comments

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.

Examples

			a(6) = 0 corresponding to a change of square radius from the open interval (3,4) to 4, i.e., the interval (A001481(3),A001481(4)) to A001481(4).
a(48) and a(49) are odd,  corresponding to the transition from (49,50) to 50 and 50 to (50,52) respectively (r = 5).
		

Links

• Rajan Murthy, Table of n, a(n) for n = 1..4999

Crossrefs

First differences of A234300.

Cf. A001481 (see comments).

Cf. A232499 (number of completely encircled squares when the radii are indexed by A000404).

Formula

a(n) = A234300(n) - A234300(n-1).