A232499 Number of unit squares, aligned with a Cartesian grid, completely within the first quadrant of a circle centered at the origin ordered by increasing radius.

1, 3, 4, 6, 8, 10, 11, 13, 15, 17, 19, 20, 22, 24, 26, 28, 30, 33, 35, 37, 39, 41, 45, 47, 48, 50, 52, 54, 56, 60, 62, 64, 66, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 89, 90, 94, 96, 98, 102, 104, 106, 108, 110, 112, 114, 115, 117, 119, 123, 125, 127, 129, 131

Offset

1,2

Comments

The interval between terms reflects the number of ways a square integer can be partitioned into the sum of two square integers in an ordered pair. As examples, the increase from a(1) to a(2) from 1 to 3 is due to the inclusion of (1,2) and (2,1); and the increase from a(2) to a(3) is due to the inclusion of (2,2). Larger intervals occur when there are more combinations, such as, between a(17) and a(18) when (1,7), (7,1), and (5,5) are included.

Links

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (terms 1..2623 from Rajan Murthy).

L. Edson Jeffery, Illustration of the first few terms.

Rajan Murthy, Graph of sequence

Rajan Murthy, Graph of intervals

Rajan Murthy, Diagram depicting A(13)

Example

When radius of the circle exceeds 2^(1/2), one square is completely within the circle until the radius reaches 5^(1/2) when three squares are completely within the circle.

Mathematica

(* An empirical solution *) terms = 100; f[r_] := Sum[Floor[Sqrt[r^2 - n^2]], {n, 1, Floor[r]}]; Clear[g]; g[m_] := g[m] = Union[Table[f[Sqrt[s]], {s, 2, m }]][[1 ;; terms]]; g[m = dm = 4*terms]; g[m = m + dm]; While[g[m] != g[m - dm], Print[m]; m = m + dm]; A232499 = g[m]  (* Jean-François Alcover, Mar 06 2014 *)

Crossrefs

First differences are in 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.

The radii corresponding to the terms are given by the square roots of A000404.

Cf. A001481, A057961.

Cf. A237707 (3-dimensional analog), A239353 (4-dimensional analog).

Sequence in context: A047298 A184748 A184742 * A246705 A300997 A024672

Adjacent sequences: A232496 A232497 A232498 * A232500 A232501 A232502

Keyword

nonn

Author

Rajan Murthy and Vale Murthy, Nov 24 2013

Status

approved