A257022 Trace of n in the quarter-sum representation of n.

0, 1, 2, 1, 4, 1, 6, 1, 2, 9, 1, 2, 12, 1, 2, 1, 16, 1, 2, 1, 20, 1, 2, 1, 4, 25, 1, 2, 1, 4, 30, 1, 2, 1, 4, 1, 36, 1, 2, 1, 4, 1, 42, 1, 2, 1, 4, 1, 6, 49, 1, 2, 1, 4, 1, 6, 56, 1, 2, 1, 4, 1, 6, 1, 64, 1, 2, 1, 4, 1, 6, 1, 72, 1, 2, 1, 4, 1, 6, 1, 2, 81

Offset: 0

Author

Clark Kimberling, Apr 15 2015

Keywords

• nonn
• easy

Comments

Every positive integer is a sum of at most four distinct quarter squares, of which the least term is the trace; see A257019.

Examples

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

Links

• Clark Kimberling, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A002620, A257019, A257020, A257021, A257023.

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[200]; r[0] = {0};
r[n_] := If[MemberQ[bb, n], {n}, Join[{g[[n]]}, r[n - g[[n]]]]];
Table[Last[r[n]], {n, 0, 3 z}] (* A257022 *)