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 1-5 of 5 results.

A308822 Sum of all the parts in the partitions of n into 5 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 5, 6, 14, 24, 45, 70, 110, 156, 234, 322, 450, 592, 799, 1026, 1330, 1680, 2121, 2618, 3243, 3936, 4800, 5746, 6885, 8148, 9657, 11310, 13237, 15360, 17820, 20502, 23590, 26928, 30747, 34884, 39546, 44600, 50266, 56364, 63167, 70488, 78615
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 26 2019

Keywords

Examples

			The partitions of n into 5 parts for n = 10, 11, ..
                                                       1+1+1+1+10
                                                        1+1+1+2+9
                                                        1+1+1+3+8
                                                        1+1+1+4+7
                                                        1+1+1+5+6
                                            1+1+1+1+9   1+1+2+2+8
                                            1+1+1+2+8   1+1+2+3+7
                                            1+1+1+3+7   1+1+2+4+6
                                            1+1+1+4+6   1+1+2+5+5
                                            1+1+1+5+5   1+1+3+3+6
                                1+1+1+1+8   1+1+2+2+7   1+1+3+4+5
                                1+1+1+2+7   1+1+2+3+6   1+1+4+4+4
                                1+1+1+3+6   1+1+2+4+5   1+2+2+2+7
                    1+1+1+1+7   1+1+1+4+5   1+1+3+3+5   1+2+2+3+6
                    1+1+1+2+6   1+1+2+2+6   1+1+3+4+4   1+2+2+4+5
                    1+1+1+3+5   1+1+2+3+5   1+2+2+2+6   1+2+3+3+5
        1+1+1+1+6   1+1+1+4+4   1+1+2+4+4   1+2+2+3+5   1+2+3+4+4
        1+1+1+2+5   1+1+2+2+5   1+1+3+3+4   1+2+2+4+4   1+3+3+3+4
        1+1+1+3+4   1+1+2+3+4   1+2+2+2+5   1+2+3+3+4   2+2+2+2+6
        1+1+2+2+4   1+1+3+3+3   1+2+2+3+4   1+3+3+3+3   2+2+2+3+5
        1+1+2+3+3   1+2+2+2+4   1+2+3+3+3   2+2+2+2+5   2+2+2+4+4
        1+2+2+2+3   1+2+2+3+3   2+2+2+2+4   2+2+2+3+4   2+2+3+3+4
        2+2+2+2+2   2+2+2+2+3   2+2+2+3+3   2+2+3+3+3   2+3+3+3+3
--------------------------------------------------------------------------
  n  |     10          11          12          13          14        ...
--------------------------------------------------------------------------
a(n) |     70         110         156         234         322        ...
--------------------------------------------------------------------------
- _Wesley Ivan Hurt_, Sep 08 2019

Links

Crossrefs

Cf. A026811, A308823, A308824, A308825, A308826, A308827.

Programs

  • Mathematica
    Table[n*Sum[Sum[Sum[Sum[1, {i, j, Floor[(n - j - k - l)/2]}], {j, k, Floor[(n - k - l)/3]}], {k, l, Floor[(n - l)/4]}], {l, Floor[n/5]}], {n, 0, 100}]

Formula

a(n) = n * A026811(n).
a(n) = A308823(n) + A308824(n) + A308825(n) + A308826(n) + A308827(n).

A308823 Sum of the smallest parts of the partitions of n into 5 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 8, 11, 15, 21, 28, 38, 48, 62, 78, 98, 122, 149, 181, 219, 262, 314, 370, 436, 510, 595, 691, 797, 916, 1050, 1198, 1365, 1545, 1747, 1968, 2212, 2480, 2771, 3089, 3437, 3814, 4227, 4669, 5151, 5670, 6232, 6838, 7487, 8185, 8936
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 26 2019

Keywords

Examples

			Figure 1: The partitions of n into 5 parts for n = 10, 11, ..
                                                       1+1+1+1+10
                                                        1+1+1+2+9
                                                        1+1+1+3+8
                                                        1+1+1+4+7
                                                        1+1+1+5+6
                                            1+1+1+1+9   1+1+2+2+8
                                            1+1+1+2+8   1+1+2+3+7
                                            1+1+1+3+7   1+1+2+4+6
                                            1+1+1+4+6   1+1+2+5+5
                                            1+1+1+5+5   1+1+3+3+6
                                1+1+1+1+8   1+1+2+2+7   1+1+3+4+5
                                1+1+1+2+7   1+1+2+3+6   1+1+4+4+4
                                1+1+1+3+6   1+1+2+4+5   1+2+2+2+7
                    1+1+1+1+7   1+1+1+4+5   1+1+3+3+5   1+2+2+3+6
                    1+1+1+2+6   1+1+2+2+6   1+1+3+4+4   1+2+2+4+5
                    1+1+1+3+5   1+1+2+3+5   1+2+2+2+6   1+2+3+3+5
        1+1+1+1+6   1+1+1+4+4   1+1+2+4+4   1+2+2+3+5   1+2+3+4+4
        1+1+1+2+5   1+1+2+2+5   1+1+3+3+4   1+2+2+4+4   1+3+3+3+4
        1+1+1+3+4   1+1+2+3+4   1+2+2+2+5   1+2+3+3+4   2+2+2+2+6
        1+1+2+2+4   1+1+3+3+3   1+2+2+3+4   1+3+3+3+3   2+2+2+3+5
        1+1+2+3+3   1+2+2+2+4   1+2+3+3+3   2+2+2+2+5   2+2+2+4+4
        1+2+2+2+3   1+2+2+3+3   2+2+2+2+4   2+2+2+3+4   2+2+3+3+4
        2+2+2+2+2   2+2+2+2+3   2+2+2+3+3   2+2+3+3+3   2+3+3+3+3
--------------------------------------------------------------------------
  n  |     10          11          12          13          14        ...
--------------------------------------------------------------------------
a(n) |      8          11          15          21          28        ...
--------------------------------------------------------------------------
- _Wesley Ivan Hurt_, Sep 08 2019

Links

Crossrefs

Cf. A026811, A308822, A308824, A308825, A308826, A308827.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[l, {i, j, Floor[(n - j - k - l)/2]}], {j, k, Floor[(n - k - l)/3]}], {k, l, Floor[(n - l)/4]}], {l, Floor[n/5]}], {n, 0, 100}]
Table[Total[IntegerPartitions[n,{5}][[;;,5]]],{n,0,60}] (* Harvey P. Dale, Nov 20 2024 *)

Formula

a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-l)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} l.
a(n) = A308822(n) - A308824(n) - A308825(n) - A308826(n) - A308827(n).
Conjectures from Colin Barker, Jun 30 2019: (Start)
G.f.: x^5 / ((1 - x)^6*(1 + x)^2*(1 + x^2)*(1 + x + x^2)*(1 + x + x^2 + x^3 + x^4)^2).
a(n) = a(n-1) + a(n-2) - 2*a(n-6) - 2*a(n-7) + a(n-8) + a(n-9) + 2*a(n-10) + a(n-11) + a(n-12) - 2*a(n-13) - 2*a(n-14) + a(n-18) + a(n-19) - a(n-20) for n>19.
(End)

A308824 Sum of the fourth largest parts in the partitions of n into 5 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 3, 6, 9, 13, 18, 27, 36, 50, 64, 86, 109, 140, 175, 220, 269, 331, 399, 486, 577, 689, 811, 959, 1119, 1305, 1508, 1747, 2003, 2300, 2617, 2984, 3376, 3821, 4300, 4839, 5415, 6060, 6749, 7521, 8337, 9243, 10207, 11273, 12404, 13641
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 26 2019

Keywords

Examples

			The partitions of n into 5 parts for n = 10, 11, ..
                                                       1+1+1+1+10
                                                        1+1+1+2+9
                                                        1+1+1+3+8
                                                        1+1+1+4+7
                                                        1+1+1+5+6
                                            1+1+1+1+9   1+1+2+2+8
                                            1+1+1+2+8   1+1+2+3+7
                                            1+1+1+3+7   1+1+2+4+6
                                            1+1+1+4+6   1+1+2+5+5
                                            1+1+1+5+5   1+1+3+3+6
                                1+1+1+1+8   1+1+2+2+7   1+1+3+4+5
                                1+1+1+2+7   1+1+2+3+6   1+1+4+4+4
                                1+1+1+3+6   1+1+2+4+5   1+2+2+2+7
                    1+1+1+1+7   1+1+1+4+5   1+1+3+3+5   1+2+2+3+6
                    1+1+1+2+6   1+1+2+2+6   1+1+3+4+4   1+2+2+4+5
                    1+1+1+3+5   1+1+2+3+5   1+2+2+2+6   1+2+3+3+5
        1+1+1+1+6   1+1+1+4+4   1+1+2+4+4   1+2+2+3+5   1+2+3+4+4
        1+1+1+2+5   1+1+2+2+5   1+1+3+3+4   1+2+2+4+4   1+3+3+3+4
        1+1+1+3+4   1+1+2+3+4   1+2+2+2+5   1+2+3+3+4   2+2+2+2+6
        1+1+2+2+4   1+1+3+3+3   1+2+2+3+4   1+3+3+3+3   2+2+2+3+5
        1+1+2+3+3   1+2+2+2+4   1+2+3+3+3   2+2+2+2+5   2+2+2+4+4
        1+2+2+2+3   1+2+2+3+3   2+2+2+2+4   2+2+2+3+4   2+2+3+3+4
        2+2+2+2+2   2+2+2+2+3   2+2+2+3+3   2+2+3+3+3   2+3+3+3+3
--------------------------------------------------------------------------
  n  |     10          11          12          13          14        ...
--------------------------------------------------------------------------
a(n) |      9          13          18          27          36        ...
--------------------------------------------------------------------------
- _Wesley Ivan Hurt_, Sep 08 2019

Links

Crossrefs

Cf. A026811, A308822, A308823, A308825, A308826, A308827.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[k, {i, j, Floor[(n - j - k - l)/2]}], {j, k, Floor[(n - k - l)/3]}], {k, l, Floor[(n - l)/4]}], {l, Floor[n/5]}], {n, 0, 100}]

Formula

a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-l)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} k.
a(n) = A308822(n) - A308823(n) - A308825(n) - A308826(n) - A308827(n).
Conjectures from Colin Barker, Jun 30 2019: (Start)
G.f.: x^5*(1 + x^3 + x^6) / ((1 - x)^6*(1 + x)^3*(1 + x^2)^2*(1 + x + x^2 + x^3 + x^4)^2).
a(n) = a(n-1) + a(n-2) - a(n-3) + 2*a(n-4) - 4*a(n-6) + a(n-8) - 3*a(n-9) + 4*a(n-10) + 4*a(n-11) - 3*a(n-12) + a(n-13) - 4*a(n-15) + 2*a(n-17) - a(n-18) + a(n-19) + a(n-20) - a(n-21) for n>20.
(End)

A308825 Sum of the third largest parts of the partitions of n into 5 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 4, 7, 11, 17, 24, 36, 50, 69, 91, 123, 158, 204, 259, 326, 403, 499, 606, 739, 886, 1060, 1256, 1489, 1745, 2041, 2371, 2750, 3166, 3643, 4160, 4750, 5393, 6112, 6897, 7774, 8720, 9772, 10910, 12168, 13518, 15006, 16601, 18352, 20229
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 26 2019

Keywords

Examples

			The partitions of n into 5 parts for n = 10, 11, ..
                                                       1+1+1+1+10
                                                        1+1+1+2+9
                                                        1+1+1+3+8
                                                        1+1+1+4+7
                                                        1+1+1+5+6
                                            1+1+1+1+9   1+1+2+2+8
                                            1+1+1+2+8   1+1+2+3+7
                                            1+1+1+3+7   1+1+2+4+6
                                            1+1+1+4+6   1+1+2+5+5
                                            1+1+1+5+5   1+1+3+3+6
                                1+1+1+1+8   1+1+2+2+7   1+1+3+4+5
                                1+1+1+2+7   1+1+2+3+6   1+1+4+4+4
                                1+1+1+3+6   1+1+2+4+5   1+2+2+2+7
                    1+1+1+1+7   1+1+1+4+5   1+1+3+3+5   1+2+2+3+6
                    1+1+1+2+6   1+1+2+2+6   1+1+3+4+4   1+2+2+4+5
                    1+1+1+3+5   1+1+2+3+5   1+2+2+2+6   1+2+3+3+5
        1+1+1+1+6   1+1+1+4+4   1+1+2+4+4   1+2+2+3+5   1+2+3+4+4
        1+1+1+2+5   1+1+2+2+5   1+1+3+3+4   1+2+2+4+4   1+3+3+3+4
        1+1+1+3+4   1+1+2+3+4   1+2+2+2+5   1+2+3+3+4   2+2+2+2+6
        1+1+2+2+4   1+1+3+3+3   1+2+2+3+4   1+3+3+3+3   2+2+2+3+5
        1+1+2+3+3   1+2+2+2+4   1+2+3+3+3   2+2+2+2+5   2+2+2+4+4
        1+2+2+2+3   1+2+2+3+3   2+2+2+2+4   2+2+2+3+4   2+2+3+3+4
        2+2+2+2+2   2+2+2+2+3   2+2+2+3+3   2+2+3+3+3   2+3+3+3+3
--------------------------------------------------------------------------
  n  |     10          11          12          13          14        ...
--------------------------------------------------------------------------
a(n) |     11          17          24          36          50        ...
--------------------------------------------------------------------------
- _Wesley Ivan Hurt_, Sep 11 2019

Links

Crossrefs

Cf. A026811, A308822, A308823, A308824, A308826, A308827.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[j, {i, j, Floor[(n - j - k - l)/2]}], {j, k, Floor[(n - k - l)/3]}], {k, l, Floor[(n - l)/4]}], {l, Floor[n/5]}], {n, 0, 100}]
Table[Total[IntegerPartitions[n,{5}][[;;,3]]],{n,0,50}] (* Harvey P. Dale, Oct 01 2024 *)

Formula

a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-l)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} j.
a(n) = A308822(n) - A308823(n) - A308824(n) - A308826(n) - A308827(n).

A308827 Sum of the largest parts of the partitions of n into 5 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 5, 9, 17, 27, 44, 64, 96, 134, 188, 251, 339, 439, 571, 724, 917, 1137, 1411, 1719, 2097, 2519, 3023, 3586, 4253, 4990, 5848, 6797, 7891, 9092, 10467, 11966, 13670, 15526, 17612, 19880, 22417, 25159, 28209, 31502, 35145, 39061, 43375
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 26 2019

Keywords

Examples

			The partitions of n into 5 parts for n = 10, 11, ..
                                                       1+1+1+1+10
                                                        1+1+1+2+9
                                                        1+1+1+3+8
                                                        1+1+1+4+7
                                                        1+1+1+5+6
                                            1+1+1+1+9   1+1+2+2+8
                                            1+1+1+2+8   1+1+2+3+7
                                            1+1+1+3+7   1+1+2+4+6
                                            1+1+1+4+6   1+1+2+5+5
                                            1+1+1+5+5   1+1+3+3+6
                                1+1+1+1+8   1+1+2+2+7   1+1+3+4+5
                                1+1+1+2+7   1+1+2+3+6   1+1+4+4+4
                                1+1+1+3+6   1+1+2+4+5   1+2+2+2+7
                    1+1+1+1+7   1+1+1+4+5   1+1+3+3+5   1+2+2+3+6
                    1+1+1+2+6   1+1+2+2+6   1+1+3+4+4   1+2+2+4+5
                    1+1+1+3+5   1+1+2+3+5   1+2+2+2+6   1+2+3+3+5
        1+1+1+1+6   1+1+1+4+4   1+1+2+4+4   1+2+2+3+5   1+2+3+4+4
        1+1+1+2+5   1+1+2+2+5   1+1+3+3+4   1+2+2+4+4   1+3+3+3+4
        1+1+1+3+4   1+1+2+3+4   1+2+2+2+5   1+2+3+3+4   2+2+2+2+6
        1+1+2+2+4   1+1+3+3+3   1+2+2+3+4   1+3+3+3+3   2+2+2+3+5
        1+1+2+3+3   1+2+2+2+4   1+2+3+3+3   2+2+2+2+5   2+2+2+4+4
        1+2+2+2+3   1+2+2+3+3   2+2+2+2+4   2+2+2+3+4   2+2+3+3+4
        2+2+2+2+2   2+2+2+2+3   2+2+2+3+3   2+2+3+3+3   2+3+3+3+3
--------------------------------------------------------------------------
  n  |     10          11          12          13          14        ...
--------------------------------------------------------------------------
a(n) |     27          44          64          96         134        ...
--------------------------------------------------------------------------
- _Wesley Ivan Hurt_, Sep 12 2019

Links

Crossrefs

Cf. A026811, A308822, A308823, A308824, A308825, A308826.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[n - i - j - k - l, {i, j, Floor[(n - j - k - l)/2]}], {j, k, Floor[(n - k - l)/3]}], {k, l, Floor[(n - l)/4]}], {l, Floor[n/5]}], {n, 0, 100}]

Formula

a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-l)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} (n-i-j-k-l).
a(n) = A308822(n) - A308823(n) - A308824(n) - A308825(n) - A308826(n).
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.