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A257058 Numbers k such that (# squares) > (# nonsquares) in the quarter-squares representation of k.

%I A257058 #6 Apr 15 2015 15:46:02
%S A257058 0,1,4,5,9,10,16,17,19,25,26,28,29,35,36,37,39,40,41,47,49,50,52,53,
%T A257058 54,61,64,65,67,68,69,71,77,81,82,84,85,86,88,95,100,101,103,104,105,
%U A257058 107,109,115,120,121,122,124,125,126,128,130,131,137,142,144,145
%N A257058 Numbers k such that (# squares) > (# nonsquares) in the quarter-squares representation of k.
%C A257058 Every positive integer is a sum of at most four distinct quarter squares; see A257019.   The sequences A257056, A257057, A257058 partition the nonnegative integers.
%H A257058 Clark Kimberling, <a href="/A257058/b257058.txt">Table of n, a(n) for n = 1..1000</a>
%e A257058 Quarter-square representations:
%e A257058 r(0) = 0, so a(1) = 0
%e A257058 r(1) = 1, so a(2) = 1
%e A257058 r(2) = 2
%e A257058 r(3) = 2 + 1
%e A257058 r(4) = 4, so a(3) = 4
%t A257058 z = 400; b[n_] := Floor[(n + 1)^2/4]; bb = Table[b[n], {n, 0, 100}];
%t A257058 s[n_] := Table[b[n], {k, b[n + 1] - b[n]}];
%t A257058 h[1] = {1}; h[n_] := Join[h[n - 1], s[n]];
%t A257058 g = h[100]; r[0] = {0};
%t A257058 r[n_] := If[MemberQ[bb, n], {n}, Join[{g[[n]]}, r[n - g[[n]]]]];
%t A257058 u = Table[Length[r[n]], {n, 0, z}]  (* A257023 *)
%t A257058 v = Table[Length[Intersection[r[n], Table[n^2, {n, 0, 1000}]]], {n, 0, z}]  (* A257024 *)
%t A257058 -1 + Select[Range[0, z], 2 v[[#]] < u[[#]] &]   (* A257056 *)
%t A257058 -1 + Select[Range[0, z], 2 v[[#]] == u[[#]] &]  (* A257057 *)
%t A257058 -1 + Select[Range[0, z], 2 v[[#]] > u[[#]] &]   (* A257058 *)
%Y A257058 Cf. A257019, A000290, A257056, A257057.
%K A257058 nonn,easy
%O A257058 1,3
%A A257058 _Clark Kimberling_, Apr 15 2015