cp's OEIS Frontend

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.

%I A229904 #23 May 01 2025 17:47:07
%S A229904 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,
%T A229904 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
%N 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.
%C A229904 From _Mohammed Yaseen_, Apr 23 2025: (Start)
%C A229904 a(n) is the number of solutions to x^2 + y^2 = A000404(n), x,y,z >= 1.
%C A229904 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)
%H A229904 Rajan Murthy, <a href="/A229904/b229904.txt">Table of n, a(n) for n = 1..2623</a>
%F A229904 a(n) = A232499(n) - A232499(n-1) for n>1, a(1) = A232499(1).
%e A229904 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)).
%Y A229904 First differences of A232499.
%Y A229904 Radii are the square roots of A000404.
%Y A229904 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.
%K A229904 nonn
%O A229904 1,2
%A A229904 _Rajan Murthy_ and _Vale Murthy_, Dec 19 2013