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A257057 Numbers k such that (# squares) = (# nonsquares) in the quarter-squares representation of k.

%I A257057 #6 Apr 15 2015 15:45:34
%S A257057 3,7,11,13,18,21,24,27,31,34,38,43,46,51,55,57,60,66,70,73,76,83,87,
%T A257057 91,94,99,102,106,111,114,119,123,127,133,136,141,146,150,157,160,165,
%U A257057 171,175,181,183,186,191,198,202,208,211,214,219,227,231,237,241
%N A257057 Numbers k such that (# squares) = (# nonsquares) in the quarter-squares representation of k.
%C A257057 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 A257057 Clark Kimberling, <a href="/A257057/b257057.txt">Table of n, a(n) for n = 1..1000</a>
%e A257057 Quarter-square representations:
%e A257057 r(0) = 0
%e A257057 r(1) = 1
%e A257057 r(2) = 2
%e A257057 r(3) = 2 + 1, so that a(1) = 3
%t A257057 z = 400; b[n_] := Floor[(n + 1)^2/4]; bb = Table[b[n], {n, 0, 100}];
%t A257057 s[n_] := Table[b[n], {k, b[n + 1] - b[n]}];
%t A257057 h[1] = {1}; h[n_] := Join[h[n - 1], s[n]];
%t A257057 g = h[100]; r[0] = {0};
%t A257057 r[n_] := If[MemberQ[bb, n], {n}, Join[{g[[n]]}, r[n - g[[n]]]]];
%t A257057 u = Table[Length[r[n]], {n, 0, z}]  (* A257023 *)
%t A257057 v = Table[Length[Intersection[r[n], Table[n^2, {n, 0, 1000}]]], {n, 0, z}]  (* A257024 *)
%t A257057 -1 + Select[Range[0, z], 2 v[[#]] < u[[#]] &]   (* A257056 *)
%t A257057 -1 + Select[Range[0, z], 2 v[[#]] == u[[#]] &]  (* A257057 *)
%t A257057 -1 + Select[Range[0, z], 2 v[[#]] > u[[#]] &]   (* A257058 *)
%Y A257057 Cf. A257019, A000290, A257056, A257058.
%K A257057 nonn,easy
%O A257057 1,1
%A A257057 _Clark Kimberling_, Apr 15 2015