A257023 Number of terms in the quarter-sum representation of n.

1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 3, 1, 2, 2, 3, 1, 2, 2, 3, 2, 1, 2, 2, 3, 2, 1, 2, 2, 3, 2, 3, 1, 2, 2, 3, 2, 3, 1, 2, 2, 3, 2, 3, 2, 1, 2, 2, 3, 2, 3, 2, 1, 2, 2, 3, 2, 3, 2, 3, 1, 2, 2, 3, 2, 3, 2, 3, 1, 2, 2, 3, 2, 3, 2, 3, 3, 1, 2, 2, 3, 2

Offset

0,4

Comments

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

Links

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

Example

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

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[Length[r[n]], {n, 0, 3 z}] (* A257022 *)

Crossrefs

Cf. A002620, A257019, A257020, A257021, A257022.

Sequence in context: A106149 A253607 A212179 * A238894 A054740 A272947

Adjacent sequences: A257020 A257021 A257022 * A257024 A257025 A257026

Keyword

nonn,easy

Author

Clark Kimberling, Apr 15 2015

Status

approved