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A257056 Numbers k such that (# squares) < (# nonsquares) in the quarter-squares representation of k.

%I A257056 #8 Apr 15 2015 15:44:55
%S A257056 2,6,8,12,14,15,20,22,23,30,32,33,42,44,45,48,56,58,59,62,63,72,74,75,
%T A257056 78,79,80,89,90,92,93,96,97,98,108,110,112,113,116,117,118,129,132,
%U A257056 134,135,138,139,140,143,152,156,158,159,162,163,164,167,168,177
%N A257056 Numbers k such that (# squares) < (# nonsquares) in the quarter-squares representation of k.
%C A257056 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 A257056 Clark Kimberling, <a href="/A257056/b257056.txt">Table of n, a(n) for n = 1..1000</a>
%e A257056 Quarter-square representations:
%e A257056 r(0) = 0
%e A257056 r(1) = 1
%e A257056 r(2) = 2, so that a(1) = 2
%e A257056 r(3) = 2 + 1
%e A257056 r(4) = 4
%e A257056 r(5) = 4 + 1
%e A257056 r(6) = 6, so that a(2) = 6
%t A257056 z = 400; b[n_] := Floor[(n + 1)^2/4]; bb = Table[b[n], {n, 0, 100}];
%t A257056 s[n_] := Table[b[n], {k, b[n + 1] - b[n]}];
%t A257056 h[1] = {1}; h[n_] := Join[h[n - 1], s[n]];
%t A257056 g = h[100]; r[0] = {0};
%t A257056 r[n_] := If[MemberQ[bb, n], {n}, Join[{g[[n]]}, r[n - g[[n]]]]];
%t A257056 u = Table[Length[r[n]], {n, 0, z}]  (* A257023 *)
%t A257056 v = Table[Length[Intersection[r[n], Table[n^2, {n, 0, 1000}]]], {n, 0, z}]  (* A257024 *)
%t A257056 -1 + Select[Range[0, z], 2 v[[#]] < u[[#]] &]   (* A257056 *)
%t A257056 -1 + Select[Range[0, z], 2 v[[#]] == u[[#]] &]  (* A257057 *)
%t A257056 -1 + Select[Range[0, z], 2 v[[#]] > u[[#]] &]   (* A257058 *)
%Y A257056 Cf. A257019, A000290, A257057, A257058.
%K A257056 nonn,easy
%O A257056 1,1
%A A257056 _Clark Kimberling_, Apr 15 2015