A257057 Numbers k such that (# squares) = (# nonsquares) in the quarter-squares representation of k.

3, 7, 11, 13, 18, 21, 24, 27, 31, 34, 38, 43, 46, 51, 55, 57, 60, 66, 70, 73, 76, 83, 87, 91, 94, 99, 102, 106, 111, 114, 119, 123, 127, 133, 136, 141, 146, 150, 157, 160, 165, 171, 175, 181, 183, 186, 191, 198, 202, 208, 211, 214, 219, 227, 231, 237, 241

Offset: 1

Author

Clark Kimberling, Apr 15 2015

Keywords

• nonn
• easy

Comments

Every positive integer is a sum of at most four distinct quarter squares; see A257019. The sequences A257056, A257057, A257058 partition the nonnegative integers.

Examples

			Quarter-square representations:
r(0) = 0
r(1) = 1
r(2) = 2
r(3) = 2 + 1, so that a(1) = 3
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A257019, A000290, A257056, A257058.

Programs

Mathematica

z = 400; b[n_] := Floor[(n + 1)^2/4]; bb = Table[b[n], {n, 0, 100}];
s[n_] := Table[b[n], {k, b[n + 1] - b[n]}];
h[1] = {1}; h[n_] := Join[h[n - 1], s[n]];
g = h[100]; r[0] = {0};
r[n_] := If[MemberQ[bb, n], {n}, Join[{g[[n]]}, r[n - g[[n]]]]];
u = Table[Length[r[n]], {n, 0, z}]  (* A257023 *)
v = Table[Length[Intersection[r[n], Table[n^2, {n, 0, 1000}]]], {n, 0, z}]  (* A257024 *)
-1 + Select[Range[0, z], 2 v[[#]] < u[[#]] &]   (* A257056 *)
-1 + Select[Range[0, z], 2 v[[#]] == u[[#]] &]  (* A257057 *)
-1 + Select[Range[0, z], 2 v[[#]] > u[[#]] &]   (* A257058 *)