A257024 - OEIS

%I #4 Apr 15 2015 15:44:12

%S 1,1,0,1,1,2,0,1,0,1,2,1,0,1,0,1,1,2,1,2,0,1,0,1,1,1,2,1,2,2,0,1,0,1,

%T 1,2,1,2,1,2,2,3,0,1,0,1,1,2,0,1,2,1,2,2,3,1,0,1,0,1,1,2,0,1,1,2,1,2,

%U 2,3,1,2,0,1,0,1,1,2,0,1,0,1,2,1,2,2

%N Number of squares in the quarter-sum representation of n.

%C Every positive integer is a sum of at most four distinct quarter squares; see A257019.

%H Clark Kimberling, <a href="/A257024/b257024.txt">Table of n, a(n) for n = 0..1000</a>

%e Quarter-square representations:

%e r(5) = 4 + 1, so a(5) = 2;

%e r(11) = 9 + 2, so a(11) = 1;

%e r(35) = 30 + 4 + 1, so a(35) = 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[100]; Take[g, 100]; r[0] = {0};

%t r[n_] := If[MemberQ[bb, n], {n}, Join[{g[[n]]}, r[n - g[[n]]]]]

%t sq = Table[n^2, {n, 0, 1000}]; t = Table[r[n], {n, 0, z}]

%t u = Table[Length[Intersection[r[n], sq]], {n, 0, 250}]

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

%K nonn,easy

%O 0,6

%A _Clark Kimberling_, Apr 15 2015