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

1,1

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.

Links

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

Example

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

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 *)

Crossrefs

Cf. A257019, A000290, A257056, A257058.

Sequence in context: A045417 A260379 A045418 * A310197 A360077 A090456

Adjacent sequences: A257054 A257055 A257056 * A257058 A257059 A257060

Keyword

nonn,easy

Author

Clark Kimberling, Apr 15 2015

Status

approved