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A257056
Numbers k such that (# squares) < (# nonsquares) in the quarter-squares representation of k.
3
2, 6, 8, 12, 14, 15, 20, 22, 23, 30, 32, 33, 42, 44, 45, 48, 56, 58, 59, 62, 63, 72, 74, 75, 78, 79, 80, 89, 90, 92, 93, 96, 97, 98, 108, 110, 112, 113, 116, 117, 118, 129, 132, 134, 135, 138, 139, 140, 143, 152, 156, 158, 159, 162, 163, 164, 167, 168, 177
OFFSET
1,1
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
r(1) = 1
r(2) = 2, so that a(1) = 2
r(3) = 2 + 1
r(4) = 4
r(5) = 4 + 1
r(6) = 6, so that a(2) = 6
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