A257019 -id:A257019 - cp's OEIS frontend

A257023 Number of terms in the quarter-sum representation of n.

1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 3, 1, 2, 2, 3, 1, 2, 2, 3, 2, 1, 2, 2, 3, 2, 1, 2, 2, 3, 2, 3, 1, 2, 2, 3, 2, 3, 1, 2, 2, 3, 2, 3, 2, 1, 2, 2, 3, 2, 3, 2, 1, 2, 2, 3, 2, 3, 2, 3, 1, 2, 2, 3, 2, 3, 2, 3, 1, 2, 2, 3, 2, 3, 2, 3, 3, 1, 2, 2, 3, 2

Offset: 0

Clark Kimberling, Apr 15 2015

Every positive integer is a sum of at most four distinct quarter squares, of which the least term is the trace; see A257019.

			Quarter-square representations:
r(0) = 0, so a(0) = 1
r(3) = 2 + 1, so a(3) = 2

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A002620, A257019, A257020, A257021, A257022.

z = 100; 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[200]; r[0] = {0};
r[n_] := If[MemberQ[bb, n], {n}, Join[{g[[n]]}, r[n - g[[n]]]]];
Table[Length[r[n]], {n, 0, 3 z}] (* A257022 *)

A257020 Numbers whose quarter-square representation consists of three terms.

Original entry on oeis.org

15, 19, 23, 28, 33, 35, 39, 41, 45, 47, 52, 54, 59, 61, 63, 67, 69, 71, 75, 77, 79, 80, 84, 86, 88, 89, 93, 95, 97, 98, 103, 105, 107, 108, 113, 115, 117, 118, 120, 124, 126, 128, 129, 131, 135, 137, 139, 140, 142, 143, 147, 149, 151, 152, 154, 155, 159, 161

Offset: 1

Clark Kimberling, Apr 15 2015

Every positive integer is a sum of at most four distinct quarter squares (see A257019).

			Quarter-square representations:
r(15) = 12 + 2 + 1, three terms; a(1) = 15
r(19) = 16 + 2 + 1, three terms; a(2) = 19

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

Cf. A002620, A257019, A257021, A257023 (trace), A257024 (number of square in quarter-square representation).

z = 100; 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, 4 z}];(* A257023 *)
Flatten[-1 + Position[u, 1]]; (* A002620 *)
Flatten[-1 + Position[u, 2]]; (* A257019 *)
Flatten[-1 + Position[u, 3]]; (* A257020 *)
Flatten[-1 + Position[u, 4]]; (* A257021 *)

A257021 Numbers whose quarter-square representation consists of four terms.

Original entry on oeis.org

255, 271, 287, 304, 321, 339, 357, 376, 395, 399, 415, 419, 435, 439, 456, 460, 477, 481, 499, 503, 521, 525, 544, 548, 567, 571, 575, 591, 595, 599, 615, 619, 623, 640, 644, 648, 665, 669, 673, 691, 695, 699, 717, 721, 725, 744, 748, 752, 771, 775, 779, 799

Offset: 1

Clark Kimberling, Apr 15 2015

Every positive integer is a sum of at most four distinct quarter squares (see A257019).

			Quarter-square representations:
r(255) = 240 + 12 + 2 + 1; four terms
r(6969) = 6889 + 72 + 6 + 2; four terms

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

Cf. A002620, A257019, A257020, A257023.

z = 100; 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, 4 z}];(* A257023 *)
Flatten[-1 + Position[u, 1]]; (* A002620 *)
Flatten[-1 + Position[u, 2]]; (* A257019 *)
Flatten[-1 + Position[u, 3]]; (* A257020 *)
Flatten[-1 + Position[u, 4]]; (* A257021 *)

A257024 Number of squares in the quarter-sum representation of n.

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 1, 2, 1, 2, 0, 1, 0, 1, 1, 1, 2, 1, 2, 2, 0, 1, 0, 1, 1, 2, 1, 2, 1, 2, 2, 3, 0, 1, 0, 1, 1, 2, 0, 1, 2, 1, 2, 2, 3, 1, 0, 1, 0, 1, 1, 2, 0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 2, 1, 2, 2

Offset: 0

Clark Kimberling, Apr 15 2015

Every positive integer is a sum of at most four distinct quarter squares; see A257019.

			Quarter-square representations:
r(5) = 4 + 1, so a(5) = 2;
r(11) = 9 + 2, so a(11) = 1;
r(35) = 30 + 4 + 1, so a(35) = 2.

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A002620, A257019, A257020, A257021, A257022.

z = 100; 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]; Take[g, 100]; r[0] = {0};
r[n_] := If[MemberQ[bb, n], {n}, Join[{g[[n]]}, r[n - g[[n]]]]]
sq = Table[n^2, {n, 0, 1000}]; t = Table[r[n], {n, 0, z}]
u = Table[Length[Intersection[r[n], sq]], {n, 0, 250}]

A257022 Trace of n in the quarter-sum representation of n.

Original entry on oeis.org

0, 1, 2, 1, 4, 1, 6, 1, 2, 9, 1, 2, 12, 1, 2, 1, 16, 1, 2, 1, 20, 1, 2, 1, 4, 25, 1, 2, 1, 4, 30, 1, 2, 1, 4, 1, 36, 1, 2, 1, 4, 1, 42, 1, 2, 1, 4, 1, 6, 49, 1, 2, 1, 4, 1, 6, 56, 1, 2, 1, 4, 1, 6, 1, 64, 1, 2, 1, 4, 1, 6, 1, 72, 1, 2, 1, 4, 1, 6, 1, 2, 81

Offset: 0

Clark Kimberling, Apr 15 2015

Every positive integer is a sum of at most four distinct quarter squares, of which the least term is the trace; see A257019.

			Quarter-square representations:
r(0) = 0, so a(0) = 0
r(1) = 1, so a(1) = 1
r(2) = 2, so a(2) = 2
r(3) = 2 + 1, so a(3) = 1

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A002620, A257019, A257020, A257021, A257023.

z = 100; 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[200]; r[0] = {0};
r[n_] := If[MemberQ[bb, n], {n}, Join[{g[[n]]}, r[n - g[[n]]]]];
Table[Last[r[n]], {n, 0, 3 z}] (* A257022 *)

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

Original entry on oeis.org

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 *)

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

Original entry on oeis.org

3, 7, 11, 13, 18, 21, 24, 27, 31, 34, 38, 43, 46, 51, 55, 57, 60, 66, 70, 73, 76, 83, 87, 91, 94, 99, 102, 106, 111, 114, 119, 123, 127, 133, 136, 141, 146, 150, 157, 160, 165, 171, 175, 181, 183, 186, 191, 198, 202, 208, 211, 214, 219, 227, 231, 237, 241

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
r(3) = 2 + 1, so that a(1) = 3

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

Cf. A257019, A000290, A257056, 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 *)

A257018 Rectangular array read by columns: row i shows the numbers whose greedy quarter-squares representation consists of i terms, for i = 1, 2, 3, 4.

Original entry on oeis.org

0, 3, 15, 255, 1, 5, 19, 271, 2, 7, 23, 287, 4, 8, 28, 304, 6, 10, 33, 321, 9, 11, 35, 339, 12, 13, 39, 357, 16, 14, 41, 376, 20, 17, 45, 395, 25, 18, 47, 399, 30, 21, 52, 415, 36, 22, 54, 419, 42, 24, 59, 435, 49, 26, 61, 439, 56, 27, 63, 456, 64, 29, 67

Offset: 1

Clark Kimberling, Apr 15 2015

Theorem: Every positive integer is a sum of at most four distinct quarter squares (proved at Math Overflow link). The greedy representation is found as follows. Let f(n) be the greatest quarter-square <= n, and apply r(n) = f(n) + r(n - f(n)) until reaching 0. The least term of r(n) is the trace of n, at A257022.

			The array:
0    1    2    4    6    9    12   ...
3    5    7    8    10   11   13   ...
15   19   23   28   33   35   39   ...
255  271  287  304  321  339  357  ...
Quarter-square representations:
r(0) = 0,
r(1) = 1,
r(2) = 2,
r(3) = 2 + 1,
r(15) = 12 + 2 + 1,
r(6969) = 6889 + 72 + 6 + 2.

Math Overflow, Every positive integer a sum of at most 4 distinct quarter-squares

Cf. A257018 (quarter-square sums), A002620 (row 1, the quarter-squares ), A257019 (row 2), A257020 (row 3); A257021 (row 4), A257023 (number of terms).

z = 200; b[n_] := Floor[(n + 1)^2/4]; bb = Table[b[n], {n, 0, z}];
s[n_] := Table[b[n], {k, b[n + 1] - b[n]}];
h[1] = {1}; h[n_] := Join[h[n - 1], s[n]]; g = h[200]; r[0] = {0};
r[n_] := If[MemberQ[bb, n], {n}, Join[{g[[n]]}, r[n - g[[n]]]]];
u = Table[Length[r[n]], {n, 0, 4 z}]  (* A257023 *)
TableForm[Table[Take[Flatten[-1 + Position[u, k]], 10], {k, 1, 4}]]  (*A257018 array *)
t = Table[Take[Flatten[-1 + Position[u, k]], 30], {k, 1, 4}];
Flatten[Table[t[[i, j]], {j, 1, 30}, {i, 1, 4}]] (*A257018 sequence *)

cp's OEIS Frontend

A257023 Number of terms in the quarter-sum representation of n.

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A257020 Numbers whose quarter-square representation consists of three terms.

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A257021 Numbers whose quarter-square representation consists of four terms.

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A257024 Number of squares in the quarter-sum representation of n.

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A257022 Trace of n in the quarter-sum representation of n.

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

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

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

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