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