%I #4 Apr 15 2015 15:43:43
%S 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,
%T 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,
%U 2,3,2,3,1,2,2,3,2,3,2,3,3,1,2,2,3,2
%N Number of terms in the quarter-sum representation of n.
%C Every positive integer is a sum of at most four distinct quarter squares, of which the least term is the trace; see A257019.
%H Clark Kimberling, <a href="/A257023/b257023.txt">Table of n, a(n) for n = 0..1000</a>
%e Quarter-square representations:
%e r(0) = 0, so a(0) = 1
%e r(3) = 2 + 1, so a(3) = 2
%t z = 100; b[n_] := Floor[(n + 1)^2/4]; bb = Table[b[n], {n, 0, 100}];
%t s[n_] := Table[b[n], {k, b[n + 1] - b[n]}];
%t h[1] = {1}; h[n_] := Join[h[n - 1], s[n]];
%t g = h[200]; r[0] = {0};
%t r[n_] := If[MemberQ[bb, n], {n}, Join[{g[[n]]}, r[n - g[[n]]]]];
%t Table[Length[r[n]], {n, 0, 3 z}] (* A257022 *)
%Y Cf. A002620, A257019, A257020, A257021, A257022.
%K nonn,easy
%O 0,4
%A _Clark Kimberling_, Apr 15 2015