cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A241086 Number of partitions p of n into distinct parts such that max(p) <= 2*(number of parts of p).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 5, 5, 6, 7, 8, 9, 11, 12, 14, 15, 17, 19, 21, 24, 27, 31, 34, 38, 42, 47, 51, 57, 62, 70, 77, 85, 93, 104, 114, 125, 137, 150, 164, 180, 196, 214, 234, 255, 279, 304, 332, 360, 393, 426, 464, 502, 545, 589, 640, 691, 749
Offset: 0

Views

Author

Clark Kimberling, Apr 17 2014

Keywords

Crossrefs

Cf. A241085, A241087, A241088, A241089.

Programs

  • Mathematica
    z = 40; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
Table[Count[f[n], p_ /; Max[p] < 2*Length[p]], {n, 0, z}]  (* A241085 *)
Table[Count[f[n], p_ /; Max[p] <= 2*Length[p]], {n, 0, z}] (* A241086 *)
Table[Count[f[n], p_ /; Max[p] == 2*Length[p]], {n, 0, z}] (* A241087 *)
Table[Count[f[n], p_ /; Max[p] >= 2*Length[p]], {n, 0, z}] (* A241088 *)
Table[Count[f[n], p_ /; Max[p] > 2*Length[p]], {n, 0, z}]  (* A241089 *)

Formula

a(15) counts these 7 partitions: 8421, 7521, 7431, 654, 6531, 6432, 54321.

A241087 Number of partitions p of n into distinct parts such that max(p) = 2*(number of parts of p).

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 2, 2, 2, 2, 2, 2, 3, 4, 4, 6, 5, 6, 6, 7, 7, 9, 10, 12, 13, 15, 16, 18, 19, 20, 23, 25, 28, 30, 35, 38, 43, 46, 51, 55, 61, 64, 72, 76, 84, 91, 101, 109, 120, 130, 142, 155, 168, 181, 196, 212, 228, 248, 266, 288, 311, 337
Offset: 0

Views

Author

Clark Kimberling, Apr 17 2014

Keywords

Examples

			a(15) counts these 2 partitions:  8421, 654.

Crossrefs

Cf. A241085, A241086, A241088, A241089.

Programs

  • Mathematica
    z = 40; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
Table[Count[f[n], p_ /; Max[p] < 2*Length[p]], {n, 0, z}]  (* A241085 *)
Table[Count[f[n], p_ /; Max[p] <= 2*Length[p]], {n, 0, z}] (* A241086 *)
Table[Count[f[n], p_ /; Max[p] == 2*Length[p]], {n, 0, z}] (* A241087 *)
Table[Count[f[n], p_ /; Max[p] >= 2*Length[p]], {n, 0, z}] (* A241088 *)
Table[Count[f[n], p_ /; Max[p] > 2*Length[p]], {n, 0, z}]  (* A241089 *)

A241085 Number of partitions p of n into distinct parts such that max(p) < 2*(number of parts of p).

Original entry on oeis.org

0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 3, 2, 3, 3, 4, 5, 6, 6, 7, 8, 8, 10, 11, 13, 14, 17, 18, 21, 22, 25, 27, 31, 33, 38, 42, 47, 52, 57, 63, 69, 76, 82, 91, 99, 109, 119, 132, 142, 158, 171, 188, 203, 223, 240, 263, 284, 309, 334, 364, 393, 428, 463, 501, 543, 588
Offset: 0

Views

Author

Clark Kimberling, Apr 17 2014

Keywords

Examples

			a(15) counts these 5 partitions:  7521, 7431, 6531, 6432, 54321.

Crossrefs

Cf. A241086, A241087, A241088, A241089.

Programs

  • Mathematica
    z = 40; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
Table[Count[f[n], p_ /; Max[p] < 2*Length[p]], {n, 0, z}]  (* A241085 *)
Table[Count[f[n], p_ /; Max[p] <= 2*Length[p]], {n, 0, z}] (* A241086 *)
Table[Count[f[n], p_ /; Max[p] == 2*Length[p]], {n, 0, z}] (* A241087 *)
Table[Count[f[n], p_ /; Max[p] >= 2*Length[p]], {n, 0, z}] (* A241088 *)
Table[Count[f[n], p_ /; Max[p] > 2*Length[p]], {n, 0, z}]  (* A241089 *)

A241088 Number of partitions p of n into distinct parts such that max(p) >= 2*(number of parts of p).

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 3, 4, 4, 6, 7, 10, 12, 15, 18, 22, 26, 32, 39, 46, 56, 66, 78, 91, 108, 125, 147, 171, 200, 231, 269, 309, 357, 410, 470, 538, 616, 703, 801, 913, 1037, 1178, 1335, 1511, 1707, 1929, 2172, 2448, 2752, 3093, 3470, 3894, 4359, 4880, 5455
Offset: 0

Views

Author

Clark Kimberling, Apr 17 2014

Keywords

Examples

			a(9) counts these 6 partitions:  9, 81, 72, 63, 621, 54.

Crossrefs

Cf. A241085, A241086, A241087, A241089.

Programs

  • Mathematica
    z = 40; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
Table[Count[f[n], p_ /; Max[p] < 2*Length[p]], {n, 0, z}]  (* A241085 *)
Table[Count[f[n], p_ /; Max[p] <= 2*Length[p]], {n, 0, z}] (* A241086 *)
Table[Count[f[n], p_ /; Max[p] == 2*Length[p]], {n, 0, z}] (* A241087 *)
Table[Count[f[n], p_ /; Max[p] >= 2*Length[p]], {n, 0, z}] (* A241088 *)
Table[Count[f[n], p_ /; Max[p] > 2*Length[p]], {n, 0, z}]  (* A241089 *)

A241091 Number of partitions p of n into distinct parts such that max(p) <= 1 + 2*(number of parts of p).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 3, 3, 3, 4, 5, 5, 7, 7, 9, 10, 11, 12, 15, 16, 19, 22, 24, 27, 30, 34, 37, 43, 47, 53, 59, 66, 72, 82, 88, 99, 109, 120, 131, 146, 160, 176, 194, 212, 233, 256, 279, 304, 334, 362, 396, 431, 471, 510, 558, 604, 659, 714, 776, 839, 913, 985
Offset: 0

Views

Author

Clark Kimberling, Apr 18 2014

Keywords

Examples

			a(12) counts these 5 partitions:  741, 732, 651, 642, 6321, 543, 5421.

Crossrefs

Cf. A241086, A241092, A241093.

Programs

  • Mathematica
    z = 30; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
 Table[Count[f[n], p_ /; Max[p] < 1 + 2*Length[p]], {n, 0, z}] (*A241086*)
 Table[Count[f[n], p_ /; Max[p] <= 1 + 2*Length[p]], {n, 0, z}](*A241091*)
 Table[Count[f[n], p_ /; Max[p] == 1 + 2*Length[p]], {n, 0, z}](*A241092*)
 Table[Count[f[n], p_ /; Max[p] >= 1 + 2*Length[p]], {n, 0, z}](*A241089*)
 Table[Count[f[n], p_ /; Max[p] > 1 + 2*Length[p]], {n, 0, z}] (*A241093*)

Formula

a(n) = A241086(n) + A241092(n) for n >= 0.

A241092 Number of partitions p of n into distinct parts such that max(p) = 1 + 2*(number of parts of p).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 5, 7, 7, 8, 9, 10, 10, 12, 13, 15, 17, 19, 21, 25, 26, 29, 32, 35, 38, 42, 46, 51, 57, 62, 69, 76, 83, 90, 100, 107, 117, 127, 139, 150, 165, 178, 195, 212, 231, 250, 273, 294, 319, 346, 373, 402
Offset: 0

Views

Author

Clark Kimberling, Apr 18 2014

Keywords

Examples

			a(12) counts these 5 partitions:  741, 732, 651, 642, 6321, 543, 5421.

Crossrefs

Cf. A241086, A241091, A241093, A000009.

Programs

  • Mathematica
    z = 30; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
Table[Count[f[n], p_ /; Max[p] < 1 + 2*Length[p]], {n, 0, z}] (*A241086*)
Table[Count[f[n], p_ /; Max[p] <= 1 + 2*Length[p]], {n, 0, z}](*A241091*)
Table[Count[f[n], p_ /; Max[p] == 1 + 2*Length[p]], {n, 0, z}](*A241092*)
Table[Count[f[n], p_ /; Max[p] >= 1 + 2*Length[p]], {n, 0, z}](*A241089*)
Table[Count[f[n], p_ /; Max[p] > 1 + 2*Length[p]], {n, 0, z}] (*A241093*)

Formula

a(n) + A241086(n) + A241093(n) = A000009(n) for n >= 1.
a(n) = A241091(n) - A241086(n) for n >= 0.

A241093 Number of partitions p of n into distinct parts such that max(p) > 1 + 2*(number of parts of p).

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 2, 3, 4, 5, 7, 8, 11, 13, 17, 21, 26, 31, 38, 45, 54, 65, 77, 92, 108, 128, 149, 175, 203, 237, 274, 318, 366, 424, 486, 559, 640, 733, 836, 953, 1084, 1232, 1398, 1583, 1792, 2025, 2286, 2576, 2902, 3262, 3666, 4111, 4610, 5160, 5774
Offset: 0

Views

Author

Clark Kimberling, Apr 18 2014

Keywords

Examples

			a(12) counts these 8 partitions: {12}, {11,1}, {10,2}, {9,3}, {9,2,1}, {8,4}, {8,3,1}, {7,5}.

Crossrefs

Cf. A241086, A241091, A241092, A000009.

Programs

  • Mathematica
    z = 30; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
Table[Count[f[n], p_ /; Max[p] < 1 + 2*Length[p]], {n, 0, z}] (*A241086*)
Table[Count[f[n], p_ /; Max[p] <= 1 + 2*Length[p]], {n, 0, z}](*A241091*)
Table[Count[f[n], p_ /; Max[p] == 1 + 2*Length[p]], {n, 0, z}](*A241092*)
Table[Count[f[n], p_ /; Max[p] >= 1 + 2*Length[p]], {n, 0, z}](*A241089*)
Table[Count[f[n], p_ /; Max[p] > 1 + 2*Length[p]], {n, 0, z}] (*A241093*)

Formula

a(n) + A241086(n) + A241093(n) = A000009(n) for n >= 1.
Showing 1-7 of 7 results.

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