A257020 Numbers whose quarter-square representation consists of three terms.

15, 19, 23, 28, 33, 35, 39, 41, 45, 47, 52, 54, 59, 61, 63, 67, 69, 71, 75, 77, 79, 80, 84, 86, 88, 89, 93, 95, 97, 98, 103, 105, 107, 108, 113, 115, 117, 118, 120, 124, 126, 128, 129, 131, 135, 137, 139, 140, 142, 143, 147, 149, 151, 152, 154, 155, 159, 161

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).

Examples

			Quarter-square representations:
r(15) = 12 + 2 + 1, three terms; a(1) = 15
r(19) = 16 + 2 + 1, three terms; a(2) = 19
		

Links

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

Crossrefs

Cf. A002620, A257019, A257021, A257023 (trace), A257024 (number of square in quarter-square representation).

Programs

Mathematica

z = 100; 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, 4 z}];(* A257023 *)
Flatten[-1 + Position[u, 1]]; (* A002620 *)
Flatten[-1 + Position[u, 2]]; (* A257019 *)
Flatten[-1 + Position[u, 3]]; (* A257020 *)
Flatten[-1 + Position[u, 4]]; (* A257021 *)