A257023 - OEIS

%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