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.

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A035634 Number of partitions of n into parts 5k+2 and 5k+3 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 1, 1, 3, 2, 3, 4, 4, 7, 6, 9, 10, 11, 16, 15, 20, 23, 25, 32, 34, 41, 47, 52, 63, 68, 80, 90, 101, 116, 129, 147, 166, 184, 210, 232, 262, 292, 326, 363, 405, 450, 501, 554, 617, 681, 756, 834, 924, 1015, 1125, 1235, 1363, 1498, 1647, 1809
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A035441-A035468, A035618-A035633, A035635-A035699.

Programs

  • Mathematica
    nmax = 61;  s1 = Range[0, nmax/5]*5 + 2; s2 = Range[0, nmax/5]*5 + 3;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 07 2020 *)
nmax = 61; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(5 k + 2)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(5 k + 3)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 16 2020 *)

Formula

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(5 k + 2)))*(-1 + 1/Product_{k>=0} (1 - x^(5 k + 3))). - Robert Price, Aug 16 2020

A035635 Number of partitions of n into parts 5k+2 and 5k+4 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 2, 2, 3, 4, 6, 4, 8, 6, 12, 10, 14, 14, 18, 21, 25, 25, 33, 33, 46, 43, 56, 57, 71, 77, 88, 95, 113, 121, 146, 148, 180, 188, 224, 238, 271, 294, 336, 364, 416, 439, 509, 540, 621, 661, 744, 805, 902, 978, 1090, 1168, 1315, 1408, 1581
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A035441-A035468, A035618-A035634, A035636-A035699.

Programs

  • Mathematica
    nmax = 64;  s1 = Range[0, nmax/5]*5 + 2; s2 = Range[0, nmax/5]*5 + 4;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 07 2020 *)
nmax = 64; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(5 k + 2)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(5 k + 4)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 16 2020 *)

Formula

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(5 k + 2)))*(-1 + 1/Product_{k>=0} (1 - x^(5 k + 4))). - Robert Price, Aug 16 2020

A035636 Number of partitions of n into parts 5k+3 and 5k+4 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 2, 1, 1, 3, 3, 4, 3, 4, 7, 7, 8, 8, 10, 14, 14, 16, 18, 20, 27, 28, 30, 35, 40, 48, 52, 55, 64, 73, 85, 90, 98, 114, 128, 143, 155, 168, 195, 214, 237, 259, 283, 319, 353, 385, 422, 460, 516, 564, 618, 672, 734, 816, 892, 964, 1057
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A035441-A035468, A035618-A035635, A035637-A035699.

Programs

  • Mathematica
    nmax = 66;  s1 = Range[0, nmax/5]*5 + 3; s2 = Range[0, nmax/5]*5 + 4;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 07 2020 *)
nmax = 66; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(5 k + 3)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(5 k + 4)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 16 2020 *)

Formula

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(5 k + 3)))*(-1 + 1/Product_{k>=0} (1 - x^(5 k + 4))). - Robert Price, Aug 16 2020

A035637 Number of partitions of n into parts 6k and 6k+1 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 4, 4, 4, 4, 4, 4, 10, 11, 11, 11, 11, 11, 22, 25, 26, 26, 26, 26, 44, 51, 54, 55, 55, 55, 84, 98, 105, 108, 109, 109, 153, 178, 193, 200, 203, 204, 270, 313, 341, 356, 363, 366, 463, 532, 582, 611, 626, 633, 774, 884, 968, 1021
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A035441-A035468, A035618-A035636, A035638-A035699.

Programs

  • Mathematica
    nmax = 64; s1 = Range[1, nmax/6]*6; s2 = Range[0, nmax/6]*6 + 1;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 07 2020 *)
nmax = 64; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(6 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(6 k + 1)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 07 2020 *)

Formula

G.f.: (-1 + 1/Product_{k>=0} (1-x^(6k+1)))*(-1 + 1/Product_{k>=1} (1-x^(6k))). - Robert Price, Aug 07 2020

A035638 Number of partitions of n into parts 6k and 6k+2 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 4, 0, 4, 0, 4, 0, 10, 0, 11, 0, 11, 0, 22, 0, 25, 0, 26, 0, 44, 0, 51, 0, 54, 0, 84, 0, 98, 0, 105, 0, 152, 0, 178, 0, 193, 0, 266, 0, 312, 0, 341, 0, 452, 0, 528, 0, 581, 0, 749, 0, 873, 0, 964, 0, 1214, 0, 1409, 0, 1561, 0, 1930, 0, 2234
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A035441-A035468, A035618-A035637, A035639-A035699.

Programs

  • Mathematica
    nmax = 76; s1 = Range[1, nmax/6]*6; s2 = Range[0, nmax/6]*6 + 2;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 12 2020 *)
nmax = 76; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(6 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(6 k + 2)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 12 2020 *)

Formula

G.f. : (-1 + 1/Product_{k>=0} (1 - x^(6 k + 2)))*(-1 + 1/Product_{k>=1} (1 - x^(6 k))). - Robert Price, Aug 12 2020

A035639 Number of partitions of n into parts 6k and 6k+3 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 4, 0, 0, 4, 0, 0, 10, 0, 0, 11, 0, 0, 22, 0, 0, 25, 0, 0, 44, 0, 0, 51, 0, 0, 83, 0, 0, 98, 0, 0, 149, 0, 0, 177, 0, 0, 259, 0, 0, 309, 0, 0, 436, 0, 0, 521, 0, 0, 716, 0, 0, 857, 0, 0, 1151, 0, 0, 1376, 0, 0, 1816, 0, 0, 2170, 0, 0, 2818, 0, 0
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A035441-A035468, A035618-A035638, A035640-A035699.
First trisection gives A006477.

Programs

  • Mathematica
    nmax = 83; s1 = Range[1, nmax/6]*6; s2 = Range[0, nmax/6]*6 + 3;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 12 2020 *)
nmax = 83; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(6 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(6 k + 3)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 12 2020 *)

Formula

G.f. : (-1 + 1/Product_{k>=0} (1 - x^(6 k + 3)))*(-1 + 1/Product_{k>=1} (1 - x^(6 k))). - Robert Price, Aug 12 2020

A035640 Number of partitions of n into parts 6k and 6k+4 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 3, 0, 1, 0, 3, 0, 7, 0, 3, 0, 8, 0, 14, 0, 8, 0, 17, 0, 26, 0, 18, 0, 33, 0, 47, 0, 36, 0, 61, 0, 81, 0, 68, 0, 106, 0, 137, 0, 121, 0, 181, 0, 224, 0, 209, 0, 296, 0, 362, 0, 347, 0, 478, 0, 570, 0, 565, 0, 750, 0, 890, 0, 894, 0, 1166, 0
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A035441-A035468, A035618-A035639, A035641-A035699.
Bisections give: A035619 (even part), A000004 (odd part).

Programs

  • Mathematica
    nmax = 81; s1 = Range[1, nmax/6]*6; s2 = Range[0, nmax/6]*6 + 4;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 12 2020 *)
nmax = 81; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(6 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(6 k + 4)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 12 2020 *)

Formula

G.f. : (-1 + 1/Product_{k>=0} (1 - x^(6 k + 4)))*(-1 + 1/Product_{k>=1} (1 - x^(6 k))). - Robert Price, Aug 12 2020

A035641 Number of partitions of n into parts 6k and 6k+5 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 3, 0, 0, 0, 1, 3, 6, 0, 0, 1, 3, 7, 11, 0, 1, 3, 7, 14, 18, 1, 3, 7, 15, 25, 30, 3, 7, 15, 28, 44, 47, 7, 15, 29, 51, 72, 73, 15, 29, 54, 87, 116, 111, 29, 55, 94, 144, 180, 167, 55, 97, 159, 230, 276, 249, 98, 166, 259, 360
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A035441-A035468, A035618-A035640, A035642-A035699.

Programs

  • Mathematica
    nmax = 75; s1 = Range[1, nmax/6]*6; s2 = Range[0, nmax/6]*6 + 5;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 12 2020 *)
nmax = 75; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(6 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(6 k + 5)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 12 2020 *)

Formula

G.f. : (-1 + 1/Product_{k>=0} (1 - x^(6 k + 5)))*(-1 + 1/Product_{k>=1} (1 - x^(6 k))). - Robert Price, Aug 12 2020

A035642 Number of partitions of n into parts 6k+1 and 6k+2 with at least one part of each type.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 3, 6, 6, 9, 9, 12, 12, 18, 19, 26, 27, 34, 35, 46, 49, 63, 66, 81, 84, 104, 111, 137, 146, 174, 183, 218, 233, 278, 297, 348, 368, 428, 457, 534, 572, 660, 702, 803, 858, 984, 1054, 1201, 1280, 1447, 1545, 1749, 1874, 2112, 2255, 2525
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A035441-A035468, A035618-A035641, A035643-A035699.

Programs

  • Maple
    b:= proc(n, i, t, s) option remember; `if`(n=0, t*s, `if`(i<1, 0,
       b(n, i-1, t, s)+(h-> `if`(h in {1, 2}, add(b(n-i*j, i-1,
      `if`(h=1, 1, t), `if`(h=2, 1, s)), j=1..n/i), 0))(irem(i, 6))))
    end:
a:= n-> b(n$2, 0$2):
seq(a(n), n=1..75);  # Alois P. Heinz, Aug 14 2020
  • Mathematica
    nmax = 57; s1 = Range[0, nmax/6]*6 + 1; s2 = Range[0, nmax/6]*6 + 2;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 13 2020 *)
nmax = 57; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(6 k + 1)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(6 k + 2)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 16 2020 *)

Formula

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(6 k + 1)))*(-1 + 1/Product_{k>=0} (1 - x^(6 k + 2))). - Robert Price, Aug 16 2020

A035643 Number of partitions of n into parts 6k+1 and 6k+3 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 2, 2, 5, 5, 5, 8, 8, 8, 14, 15, 15, 22, 23, 23, 34, 37, 38, 51, 54, 55, 74, 81, 84, 108, 116, 119, 151, 165, 172, 213, 230, 238, 290, 317, 332, 399, 433, 451, 535, 583, 613, 720, 781, 818, 950, 1033, 1088, 1257, 1363, 1432, 1638, 1777, 1875
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A035441-A035468, A035618-A035642, A035644-A035699.

Programs

  • Mathematica
    nmax = 60; s1 = Range[0, nmax/6]*6 + 1; s2 = Range[0, nmax/6]*6 + 3;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 13 2020 *)
nmax = 60; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(6 k + 1)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(6 k + 3)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 16 2020 *)

Formula

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(6 k + 1)))*(-1 + 1/Product_{k>=0} (1 - x^(6 k + 3))). - Robert Price, Aug 16 2020
Previous Showing 31-40 of 83 results. Next

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