A235141 - OEIS

%I #18 Feb 26 2018 19:17:55

%S 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,

%T -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,

%U 0,2,-4,4,-2,2,-1,1,-2,2,-2,2,-2,2,0,2,-2,2,-4,4,-2,2,-2,2

%N First differences of A234300.

%C 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.

%C 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.

%C 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.

%H Rajan Murthy, <a href="/A235141/b235141.txt">Table of n, a(n) for n = 1..4999</a>

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

%e 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).

%e a(48) and a(49) are odd,  corresponding to the transition from (49,50) to 50 and 50 to (50,52) respectively (r = 5).

%Y First differences of A234300.

%Y Cf. A001481 (see comments).

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

%K sign

%O 1,3

%A _Rajan Murthy_, Jan 03 2014