A257058 Numbers k such that (# squares) > (# nonsquares) in the quarter-squares representation of k.

0, 1, 4, 5, 9, 10, 16, 17, 19, 25, 26, 28, 29, 35, 36, 37, 39, 40, 41, 47, 49, 50, 52, 53, 54, 61, 64, 65, 67, 68, 69, 71, 77, 81, 82, 84, 85, 86, 88, 95, 100, 101, 103, 104, 105, 107, 109, 115, 120, 121, 122, 124, 125, 126, 128, 130, 131, 137, 142, 144, 145

Offset

1,3

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, so a(1) = 0
r(1) = 1, so a(2) = 1
r(2) = 2
r(3) = 2 + 1
r(4) = 4, so a(3) = 4

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, A257057.

Sequence in context: A191888 A050036 A059582 * A189889 A329451 A308578

Adjacent sequences: A257055 A257056 A257057 * A257059 A257060 A257061

Keyword

nonn,easy

Author

Clark Kimberling, Apr 15 2015

Status

approved