A257057 -id:A257057 - cp's OEIS frontend

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

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

Clark Kimberling, Apr 15 2015

Every positive integer is a sum of at most four distinct quarter squares; see A257019. The sequences A257056, A257057, A257058 partition the nonnegative integers.

			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

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A257019, A000290, A257057, A257058.

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 *)

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

Original entry on oeis.org

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

Clark Kimberling, Apr 15 2015

Every positive integer is a sum of at most four distinct quarter squares; see A257019. The sequences A257056, A257057, A257058 partition the nonnegative integers.

			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

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A257019, A000290, A257056, A257057.

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 *)

cp's OEIS Frontend

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

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