A211339 Number of integer pairs (x,y) such that 1 < x <= y <= n and x^2 + y^2 <= n.

0, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 7, 7, 8, 8, 8, 8, 8, 9, 10, 10, 10, 11, 11, 11, 12, 12, 13, 13, 13, 14, 14, 14, 15, 16, 16, 16, 16, 17, 17, 17, 17, 17, 19, 19, 20, 21, 21, 21, 21, 21, 22, 22, 22, 23, 23, 23, 23, 25, 25, 25, 26, 26, 26, 26, 27, 28, 29

Offset: 1

Author

Clark Kimberling, Apr 08 2012

Keywords

• nonn

Comments

Partial sums of A025426.

For a guide to related sequences, see A211266.

Crossrefs

Cf. A211266.

Programs

Mathematica

a = 1; b = n; z1 = 120;
t[n_] := t[n] = Flatten[Table[x^2 + y^2, {x, a, b - 1}, {y, x, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
TableForm[Table[c[n, k], {n, 1, 10}, {k, 1, 2 n}]]
Table[c[n, n], {n, 1, z1}]   (* A025426 *)
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, a, m}]
Table[c1[n, n], {n, 1, z1}]  (* A211339 *)

Formula

a(n) = -1/2(-1 + floor(sqrt(n/2)))(floor(sqrt(n/2))) + Sum_{k=1..floor(sqrt(n/2))} floor(sqrt(n - k^2)). - Nicholas Stearns, Apr 03 2017