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 1-3 of 3 results.

A304705 Number of partitions (d1,d2,...,dm) of n such that d1/1 >= d2/2 >= ... >= dm/m and 0 < d1 <= d2 <= ... <= dm.

Original entry on oeis.org

1, 1, 2, 3, 3, 4, 6, 5, 6, 8, 9, 9, 12, 11, 14, 17, 16, 17, 23, 22, 27, 31, 30, 33, 40, 41, 46, 50, 54, 57, 70, 70, 77, 88, 92, 99, 111, 115, 129, 142, 152, 160, 175, 183, 199, 223, 234, 255, 283, 299, 328, 347, 370, 390, 430, 455, 489, 523, 557, 592, 642, 674, 724, 784
Offset: 0

Views

Author

Seiichi Manyama, May 17 2018

Keywords

Examples

			n | Partition (d1,d2,...,dm)    | (d1/1, d2/2, ... , dm/m)
--+-----------------------------+---------------------------------------------
1 | (1)                         | (1)
2 | (2)                         | (2)
  | (1, 1)                      | (1, 1/2)
3 | (3)                         | (3)
  | (1, 2)                      | (1, 1)
  | (1, 1, 1)                   | (1, 1/2, 1/3)
4 | (4)                         | (4)
  | (2, 2)                      | (2, 1)
  | (1, 1, 1, 1)                | (1, 1/2, 1/3, 1/4)
5 | (5)                         | (5)
  | (2, 3)                      | (2, 3/2)
  | (1, 2, 2)                   | (1, 1, 2/3)
  | (1, 1, 1, 1, 1)             | (1, 1/2, 1/3, 1/4, 1/5)
6 | (6)                         | (6)
  | (2, 4)                      | (2, 2)
  | (3, 3)                      | (3, 3/2)
  | (1, 2, 3)                   | (1, 1, 1)
  | (2, 2, 2)                   | (2, 1, 2/3)
  | (1, 1, 1, 1, 1, 1)          | (1, 1/2, 1/3, 1/4, 1/5, 1/6)
7 | (7)                         | (7)
  | (3, 4)                      | (3, 2)
  | (2, 2, 3)                   | (2, 1, 1)
  | (1, 2, 2, 2)                | (1, 1, 2/3, 1/2)
  | (1, 1, 1, 1, 1, 1, 1)       | (1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7)
8 | (8)                         | (8)
  | (3, 5)                      | (3, 5/2)
  | (4, 4)                      | (4, 2/1)
  | (2, 3, 3)                   | (2, 3/2, 1)
  | (2, 2, 2, 2)                | (2, 1, 2/3, 1/2)
  | (1, 1, 1, 1, 1, 1, 1, 1)    | (1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8)
9 | (9)                         | (9)
  | (3, 6)                      | (3, 3)
  | (4, 5)                      | (4, 5/2)
  | (2, 3, 4)                   | (2, 3/2, 4/3)
  | (3, 3, 3)                   | (3, 3/2, 1)
  | (1, 2, 3, 3)                | (1, 1, 1, 3/4)
  | (1, 2, 2, 2, 2)             | (1, 1, 2/3, 1/2, 2/5)
  | (1, 1, 1, 1, 1, 1, 1, 1, 1) | (1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9)

Links

Crossrefs

Cf. A053251, A053282, A304706, A304707, A304708.

Programs

  • Maple
    b:= proc(n, r, i, t) option remember; `if`(n=0, 1, `if`(i>n, 0,
      b(n, r, i+1, t)+`if`(i/t>r, 0, b(n-i, i/t, i, t+1))))
    end:
a:= n-> b(n$2, 1$2):
seq(a(n), n=0..80);  # Alois P. Heinz, May 17 2018
  • Mathematica
    b[n_, r_, i_, t_] := b[n, r, i, t] = If[n == 0, 1, If[i > n, 0, b[n, r, i + 1, t] + If[i/t > r, 0, b[n - i, i/t, i, t + 1]]]];
a[n_] := b[n, n, 1, 1];
a /@ Range[0, 80] (* Jean-François Alcover, Nov 23 2020, after Alois P. Heinz *)

A304706 Number of partitions (d1,d2,...,dm) of n such that d1/1 > d2/2 > ... > dm/m and 0 < d1 <= d2 <= ... <= dm.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 3, 6, 5, 6, 6, 8, 7, 11, 10, 11, 12, 15, 14, 18, 17, 20, 23, 27, 25, 31, 32, 35, 38, 43, 43, 51, 54, 59, 63, 71, 73, 85, 89, 96, 102, 113, 120, 134, 141, 149, 161, 175, 183, 203, 213, 233, 252, 280, 293, 319, 338, 360, 383, 409, 430, 468, 493, 531, 565
Offset: 0

Views

Author

Seiichi Manyama, May 17 2018

Keywords

Examples

			n | Partition (d1,d2,...,dm)    | (d1/1, d2/2, ... , dm/m)
--+-----------------------------+---------------------------------------------
1 | (1)                         | (1)
2 | (2)                         | (2)
  | (1, 1)                      | (1, 1/2)
3 | (3)                         | (3)
  | (1, 1, 1)                   | (1, 1/2, 1/3)
4 | (4)                         | (4)
  | (2, 2)                      | (2, 1)
  | (1, 1, 1, 1)                | (1, 1/2, 1/3, 1/4)
5 | (5)                         | (5)
  | (2, 3)                      | (2, 3/2)
  | (1, 1, 1, 1, 1)             | (1, 1/2, 1/3, 1/4, 1/5)
6 | (6)                         | (6)
  | (3, 3)                      | (3, 3/2)
  | (2, 2, 2)                   | (2, 1, 2/3)
  | (1, 1, 1, 1, 1, 1)          | (1, 1/2, 1/3, 1/4, 1/5, 1/6)
7 | (7)                         | (7)
  | (3, 4)                      | (3, 2)
  | (1, 1, 1, 1, 1, 1, 1)       | (1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7)
8 | (8)                         | (8)
  | (3, 5)                      | (3, 5/2)
  | (4, 4)                      | (4, 2/1)
  | (2, 3, 3)                   | (2, 3/2, 1)
  | (2, 2, 2, 2)                | (2, 1, 2/3, 1/2)
  | (1, 1, 1, 1, 1, 1, 1, 1)    | (1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8)
9 | (9)                         | (9)
  | (4, 5)                      | (4, 5/2)
  | (2, 3, 4)                   | (2, 3/2, 4/3)
  | (3, 3, 3)                   | (3, 3/2, 1)
  | (1, 1, 1, 1, 1, 1, 1, 1, 1) | (1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9)

Crossrefs

Cf. A053251, A053282, A304705, A304707, A304708.

Programs

  • Maple
    b:= proc(n, r, i, t) option remember; `if`(n=0, 1, `if`(i>n, 0,
      b(n, r, i+1, t)+`if`(i/t>=r, 0, b(n-i, i/t, i, t+1))))
    end:
a:= n-> b(n, n+1, 1$2):
seq(a(n), n=0..80);  # Alois P. Heinz, May 17 2018
  • Mathematica
    b[n_, r_, i_, t_] := b[n, r, i, t] = If[n == 0, 1, If[i > n, 0, b[n, r, i + 1, t] + If[i/t >= r, 0, b[n - i, i/t, i, t + 1]]]];
a[n_] := b[n, n + 1, 1, 1];
a /@ Range[0, 80] (* Jean-François Alcover, Nov 23 2020, after Alois P. Heinz *)

Formula

a(n) <= A304705(n).

A304707 Number of partitions (d1,d2,...,dm) of n such that d1/1 >= d2/2 >= ... >= dm/m and d1 < d2 < ... < dm.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 2, 2, 4, 3, 4, 5, 4, 5, 7, 5, 7, 8, 8, 10, 12, 10, 11, 14, 14, 14, 18, 17, 20, 23, 22, 26, 30, 29, 32, 35, 34, 37, 43, 44, 48, 54, 54, 59, 67, 70, 76, 81, 84, 89, 97, 101, 110, 119, 123, 129, 139, 145, 155, 171, 176, 189, 201, 211, 228, 245, 257, 274, 295
Offset: 0

Views

Author

Seiichi Manyama, May 17 2018

Keywords

Examples

			n | Partition (d1,d2,...,dm)    | (d1/1, d2/2, ... , dm/m)
--+-----------------------------+-------------------------
1 | (1)                         | (1)
2 | (2)                         | (2)
3 | (3)                         | (3)
  | (1, 2)                      | (1, 1)
4 | (4)                         | (4)
5 | (5)                         | (5)
  | (2, 3)                      | (2, 3/2)
6 | (6)                         | (6)
  | (2, 4)                      | (2, 2)
  | (1, 2, 3)                   | (1, 1, 1)
7 | (7)                         | (7)
  | (3, 4)                      | (3, 2)
8 | (8)                         | (8)
  | (3, 5)                      | (3, 5/2)
9 | (9)                         | (9)
  | (3, 6)                      | (3, 3)
  | (4, 5)                      | (4, 5/2)
  | (2, 3, 4)                   | (2, 3/2, 4/3)

Crossrefs

Cf. A053251, A053282, A304705, A304706, A304708.

Programs

  • Maple
    b:= proc(n, r, i, t) option remember; `if`(n=0, 1, `if`(i>n, 0,
      b(n, r, i+1, t) +`if`(i/t>r, 0, b(n-i, i/t, i+1, t+1))))
    end:
a:= n-> b(n$2, 1$2):
seq(a(n), n=0..80);  # Alois P. Heinz, May 17 2018
  • Mathematica
    b[n_, r_, i_, t_] := b[n, r, i, t] = If[n == 0, 1, If[i > n, 0, b[n, r, i + 1, t] + If[i/t > r, 0, b[n - i, i/t, i + 1, t + 1]]]];
a[n_] := b[n, n, 1, 1];
a /@ Range[0, 80] (* Jean-François Alcover, Nov 23 2020, after Alois P. Heinz *)
Showing 1-3 of 3 results.

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