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 1-8 of 8 results.

A241740 Number of partitions p of n such that (number of numbers in p of form 3k+2) < (number of numbers in p of form 3k).

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 3, 4, 4, 7, 10, 12, 17, 24, 30, 40, 53, 70, 90, 118, 152, 194, 244, 316, 396, 497, 626, 784, 960, 1202, 1483, 1816, 2230, 2738, 3312, 4042, 4908, 5922, 7141, 8627, 10327, 12388, 14832, 17703, 21075, 25120, 29795, 35321, 41822, 49439, 58286
Offset: 0

Views

Author

Clark Kimberling, Apr 28 2014

Keywords

Comments

Each number in p is counted once, regardless of its multiplicity.

Examples

			a(8) counts these 4 partitions:  611, 431, 3311, 311111.

Crossrefs

Cf. A241737, A241741, A241741, A241743.

Programs

  • Mathematica
    z = 40; f[n_] := f[n] = IntegerPartitions[n]; s[k_, p_] := Count[Mod[DeleteDuplicates[p], 3], k];
Table[Count[f[n], p_ /; s[2, p] < s[0, p]], {n, 0, z}]  (* A241740 *)
Table[Count[f[n], p_ /; s[2, p] == s[0, p]], {n, 0, z}] (* A241741 *)
Table[Count[f[n], p_ /; s[2, p] > s[0, p]], {n, 0, z}]  (* A241742 *)

Formula

a(n) + A241741(n) + A241842(n) = A000041(n) for n >= 0.

A241743 Number of partitions p of n such that (number of numbers in p of form 3k) < (number of numbers in p of form 3k+1).

Original entry on oeis.org

0, 1, 1, 2, 3, 4, 6, 8, 12, 16, 21, 30, 40, 52, 72, 91, 121, 159, 202, 260, 335, 421, 535, 674, 840, 1052, 1304, 1614, 1996, 2451, 3002, 3674, 4468, 5442, 6592, 7971, 9624, 11584, 13898, 16691, 19947, 23823, 28410, 33782, 40113, 47610, 56302, 66572, 78569
Offset: 0

Views

Author

Clark Kimberling, Apr 28 2014

Keywords

Comments

Each number in p is counted once, regardless of its multiplicity.

Examples

			a(8) counts these 12 partitions: 71, 521, 5111, 44, 431, 422, 4211, 41111, 22211, 221111, 2111111, 11111111.

Links

Crossrefs

Cf. A241737, A241740, A241744, A241745.

Programs

  • Mathematica
    z = 40; f[n_] := f[n] = IntegerPartitions[n]; s[k_, p_] := Count[Mod[DeleteDuplicates[p], 3], k];
Table[Count[f[n], p_ /; s[0, p] < s[2, p]], {n, 0, z}]  (* A241743 *)
Table[Count[f[n], p_ /; s[0, p] == s[1, p]], {n, 0, z}] (* A241744 *)
Table[Count[f[n], p_ /; s[0, p] > s[1, p]], {n, 0, z}]  (* A241745 *)

Formula

a(n) + A241744(n) + A241845(n) = A000041(n) for n >= 0.

A241738 Number of partitions p of n such that (number of numbers in p of form 3k+1) = (number of numbers in p of form 3k+2).

Original entry on oeis.org

1, 0, 0, 2, 1, 2, 7, 5, 7, 17, 14, 18, 39, 32, 42, 76, 71, 88, 157, 143, 182, 293, 292, 357, 562, 558, 692, 1023, 1060, 1286, 1854, 1932, 2347, 3246, 3464, 4153, 5639, 6030, 7207, 9526, 10324, 12240, 15912, 17311, 20444, 26104, 28585, 33567, 42326, 46469
Offset: 0

Views

Author

Clark Kimberling, Apr 28 2014

Keywords

Comments

Each number in p is counted once, regardless of its multiplicity.

Examples

			a(8) counts these 7 partitions:  5111, 422, 3221, 3211, 22211, 221111, 2111111.

Crossrefs

Cf. A241737, A241739, A241740, A241743.

Programs

  • Mathematica
    z = 40; f[n_] := f[n] = IntegerPartitions[n]; s[k_, p_] := Count[Mod[DeleteDuplicates[p], 3], k];
Table[Count[f[n], p_ /; s[1, p] < s[2, p]], {n, 0, z}]  (* A241737 *)
Table[Count[f[n], p_ /; s[1, p] == s[2, p]], {n, 0, z}] (* A241738 *)
Table[Count[f[n], p_ /; s[1, p] > s[2, p]], {n, 0, z}]  (* A241739 *)

Formula

a(n) + A241737(n) + A241839(n) = A000041(n) for n >= 0.

A241739 Number of partitions p of n such that (number of numbers in p of form 3k+1) > (number of numbers in p of form 3k+2).

Original entry on oeis.org

0, 1, 1, 1, 3, 3, 3, 8, 9, 10, 20, 24, 27, 49, 58, 69, 109, 132, 153, 234, 279, 331, 469, 565, 662, 918, 1093, 1290, 1723, 2056, 2411, 3165, 3751, 4411, 5656, 6700, 7839, 9932, 11707, 13699, 17084, 20099, 23441, 28939, 33914, 39498, 48236, 56392, 65481
Offset: 0

Views

Author

Clark Kimberling, Apr 28 2014

Keywords

Comments

Each number in p is counted once, regardless of its multiplicity.

Examples

			a(8) counts these 9 partitions: 71, 611, 44, 431, 4211, 3311, 311111, 11111111.

Crossrefs

Cf. A241737, A241738, A241740, A241743.

Programs

  • Mathematica
    z = 40; f[n_] := f[n] = IntegerPartitions[n]; s[k_, p_] := Count[Mod[DeleteDuplicates[p], 3], k];
Table[Count[f[n], p_ /; s[1, p] < s[2, p]], {n, 0, z}]  (* A241737 *)
Table[Count[f[n], p_ /; s[1, p] == s[2, p]], {n, 0, z}] (* A241738 *)
Table[Count[f[n], p_ /; s[1, p] > s[2, p]], {n, 0, z}]  (* A241739 *)

Formula

a(n) + A241737(n) + A241838(n) = A000041(n) for n >= 0.

A241741 Number of partitions p of n such that (number of numbers in p of form 3k+2) = (number of numbers in p of form 3k).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 3, 5, 9, 11, 14, 22, 29, 36, 51, 66, 83, 107, 139, 170, 216, 273, 340, 415, 520, 635, 778, 952, 1177, 1414, 1724, 2094, 2527, 3038, 3691, 4411, 5286, 6345, 7586, 9008, 10778, 12796, 15163, 17979, 21288, 25059, 29608, 34861, 40927, 48035
Offset: 0

Views

Author

Clark Kimberling, Apr 28 2014

Keywords

Comments

Each number in p is counted once, regardless of its multiplicity.

Examples

			a(8) counts these 9 partitions:  71, 62, 53, 44, 41111, 332, 3221, 32111, 11111111.

Crossrefs

Cf. A241737, A241740, A241742, A241743.

Programs

  • Mathematica
    z = 40; f[n_] := f[n] = IntegerPartitions[n]; s[k_, p_] := Count[Mod[DeleteDuplicates[p], 3], k];
Table[Count[f[n], p_ /; s[2, p] < s[0, p]], {n, 0, z}]  (* A241740 *)
Table[Count[f[n], p_ /; s[2, p] == s[0, p]], {n, 0, z}] (* A241741 *)
Table[Count[f[n], p_ /; s[2, p] > s[0, p]], {n, 0, z}]  (* A241742 *)

Formula

a(n) + A241740(n) + A241842(n) = A000041(n) for n >= 0.

A241742 Number of partitions p of n such that (number of numbers in p of form 3k+2) > (number of numbers in p of form 3k).

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 5, 6, 9, 12, 18, 22, 31, 41, 54, 70, 95, 120, 156, 202, 259, 325, 418, 524, 659, 826, 1032, 1274, 1581, 1949, 2397, 2932, 3592, 4367, 5307, 6430, 7783, 9370, 11288, 13550, 16233, 19399, 23179, 27579, 32812, 38955, 46155, 54572, 64524, 76051
Offset: 0

Views

Author

Clark Kimberling, Apr 28 2014

Keywords

Comments

Each number in p is counted once, regardless of its multiplicity.

Examples

			a(8) counts these 9 partitions:  8, 521, 5111, 422, 4211, 2222, 22211, 221111, 2111111.

Crossrefs

Cf. A241737, A241740, A241741, A241743.

Programs

  • Mathematica
    z = 40; f[n_] := f[n] = IntegerPartitions[n]; s[k_, p_] := Count[Mod[DeleteDuplicates[p], 3], k];
Table[Count[f[n], p_ /; s[2, p] < s[0, p]], {n, 0, z}]  (* A241740 *)
Table[Count[f[n], p_ /; s[2, p] == s[0, p]], {n, 0, z}] (* A241741 *)
Table[Count[f[n], p_ /; s[2, p] > s[0, p]], {n, 0, z}]  (* A241742 *)

Formula

a(n) + A241740(n) + A241841(n) = A000041(n) for n >= 0.

A241744 Number of partitions p of n such that (number of numbers in p of form 3k) = (number of numbers in p of form 3k+1).

Original entry on oeis.org

1, 0, 1, 0, 2, 2, 3, 6, 7, 10, 17, 18, 27, 36, 44, 61, 76, 93, 124, 151, 193, 241, 297, 369, 462, 558, 707, 850, 1044, 1281, 1561, 1884, 2323, 2761, 3367, 4050, 4857, 5826, 7024, 8307, 9982, 11840, 14058, 16684, 19785, 23265, 27585, 32379, 38125, 44760
Offset: 1

Views

Author

Clark Kimberling, Apr 28 2014

Keywords

Comments

Each number in p is counted once, regardless of its multiplicity.

Examples

			a(8) counts these 7 partitions:  8, 611, 3311, 3221, 32111, 311111, 2222.

Crossrefs

Cf. A241737, A241740, A241743, A241745.

Programs

  • Mathematica
    z = 40; f[n_] := f[n] = IntegerPartitions[n]; s[k_, p_] := Count[Mod[DeleteDuplicates[p], 3], k];
Table[Count[f[n], p_ /; s[0, p] < s[2, p]], {n, 0, z}]  (* A241743 *)
Table[Count[f[n], p_ /; s[0, p] == s[1, p]], {n, 0, z}] (* A241744 *)
Table[Count[f[n], p_ /; s[0, p] > s[1, p]], {n, 0, z}]  (* A241745 *)

Formula

a(n) + A241744(n) + A241845(n) = A000041(n) for n >= 0.

A241745 Number of partitions p of n such that (number of numbers in p of form 3k) > (number of numbers in p of form 3k+1).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 2, 1, 3, 4, 4, 8, 10, 13, 19, 24, 34, 45, 59, 79, 99, 130, 170, 212, 273, 348, 425, 546, 678, 833, 1041, 1284, 1558, 1940, 2351, 2862, 3496, 4227, 5093, 6187, 7409, 8920, 10706, 12795, 15277, 18259, 21671, 25803, 30579, 36218, 42836, 50596
Offset: 0

Views

Author

Clark Kimberling, Apr 28 2014

Keywords

Comments

Each number in p is counted once, regardless of its multiplicity.

Examples

			a(8) counts these 3 partitions:  62, 53, 332.

Crossrefs

Cf. A241737, A241740, A241743, A241744.

Programs

  • Mathematica
    z = 40; f[n_] := f[n] = IntegerPartitions[n]; s[k_, p_] := Count[Mod[DeleteDuplicates[p], 3], k];
Table[Count[f[n], p_ /; s[0, p] < s[2, p]], {n, 0, z}]  (* A241743 *)
Table[Count[f[n], p_ /; s[0, p] == s[1, p]], {n, 0, z}] (* A241744 *)
Table[Count[f[n], p_ /; s[0, p] > s[1, p]], {n, 0, z}]  (* A241745 *)

Formula

a(n) + A241744(n) + A241845(n) = A000041(n) for n >= 0.
Showing 1-8 of 8 results.

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