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

0,3

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

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 *)

Crossrefs

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

Sequence in context: A295127 A076512 A128707 * A214721 A324384 A340081

Adjacent sequences: A257019 A257020 A257021 * A257023 A257024 A257025

Keyword

nonn,easy

Author

Clark Kimberling, Apr 15 2015

Status

approved