A229904 Number of additional unit squares completely encircled in the first quadrant of a Cartesian grid by a circle centered at the origin as the radius squared increases from one sum of two square integers to the next larger sum of two square integers.

1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 4, 2, 1, 2, 2, 2, 2, 4, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 1, 4, 2, 2, 4, 2, 2, 2, 2, 2, 2, 1, 2, 2, 4, 2, 2, 2, 2

Offset: 1

Author

Rajan Murthy and Vale Murthy, Dec 19 2013

Keywords

• nonn

Comments

From Mohammed Yaseen, Apr 23 2025: (Start)

a(n) is the number of solutions to x^2 + y^2 = A000404(n), x,y,z >= 1.

a(n) are the degeneracies of the energy levels of a particle in a two-dimensional box in quantum mechanics. See A014465 for the three-dimensional box case. (End)

Examples

			When the radius increases from 0 to sqrt(2), one square is completely encircled (a(1)).  When the radius increases from sqrt(2) to sqrt(3), two more squares are encircled (a(2)).  When the radius increases from sqrt(45) to sqrt(50), three more squares are encircled(a(18)).
		

Links

• Rajan Murthy, Table of n, a(n) for n = 1..2623

Crossrefs

First differences of A232499.

Radii are the square roots of A000404.

The first differences must be odd at positions given in A024517 by proof by symmetry as r^2=2*n^2 is on the x=y line.

Formula

a(n) = A232499(n) - A232499(n-1) for n>1, a(1) = A232499(1).