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 1-4 of 4 results.

A308733 Sum of the smallest parts of the partitions of n into 4 parts.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 3, 6, 7, 11, 14, 21, 25, 34, 41, 55, 64, 81, 95, 119, 136, 165, 189, 227, 256, 301, 339, 396, 441, 507, 564, 645, 711, 804, 885, 996, 1089, 1215, 1326, 1474, 1600, 1766, 1914, 2106, 2272, 2486, 2678, 2922, 3136, 3406, 3650, 3955, 4225, 4560
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 22 2019

Keywords

Examples

			Figure 1: The partitions of n into 4 parts for n = 8, 9, ..
                                                         1+1+1+9
                                                         1+1+2+8
                                                         1+1+3+7
                                                         1+1+4+6
                                             1+1+1+8     1+1+5+5
                                             1+1+2+7     1+2+2+7
                                 1+1+1+7     1+1+3+6     1+2+3+6
                                 1+1+2+6     1+1+4+5     1+2+4+5
                                 1+1+3+5     1+2+2+6     1+3+3+5
                     1+1+1+6     1+1+4+4     1+2+3+5     1+3+4+4
         1+1+1+5     1+1+2+5     1+2+2+5     1+2+4+4     2+2+2+6
         1+1+2+4     1+1+3+4     1+2+3+4     1+3+3+4     2+2+3+5
         1+1+3+3     1+2+2+4     1+3+3+3     2+2+2+5     2+2+4+4
         1+2+2+3     1+2+3+3     2+2+2+4     2+2+3+4     2+3+3+4
         2+2+2+2     2+2+2+3     2+2+3+3     2+3+3+3     3+3+3+3
--------------------------------------------------------------------------
  n  |      8           9          10          11          12        ...
--------------------------------------------------------------------------
a(n) |      6           7          11          14          21        ...
--------------------------------------------------------------------------
- _Wesley Ivan Hurt_, Sep 07 2019

Links

Crossrefs

Cf. A026810, A308758, A308759, A308760, A308775.

Programs

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

Formula

a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} k.
a(n) = A308775(n) - A308758(n) - A308759(n) - A308760(n).
Conjectures from Colin Barker, Jun 23 2019: (Start)
G.f.: x^4 / ((1 - x)^5*(1 + x)^3*(1 + x^2)^2*(1 + x + x^2)).
a(n) = a(n-1) + a(n-2) + a(n-4) - 3*a(n-5) - a(n-6) + a(n-8) + 3*a(n-9) - a(n-10) - a(n-12) - a(n-13) + a(n-14) for n>13.
(End)

A308759 Sum of the second largest parts of the partitions of n into 4 parts.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 3, 5, 10, 13, 23, 30, 46, 59, 83, 103, 141, 170, 220, 265, 334, 392, 484, 563, 680, 784, 930, 1061, 1247, 1409, 1631, 1836, 2106, 2349, 2673, 2967, 3348, 3699, 4143, 4554, 5077, 5554, 6150, 6710, 7396, 8032, 8816, 9546, 10432, 11264, 12260
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 22 2019

Keywords

Examples

			Figure 1: The partitions of n into 4 parts for n = 8, 9, ..
                                                         1+1+1+9
                                                         1+1+2+8
                                                         1+1+3+7
                                                         1+1+4+6
                                             1+1+1+8     1+1+5+5
                                             1+1+2+7     1+2+2+7
                                 1+1+1+7     1+1+3+6     1+2+3+6
                                 1+1+2+6     1+1+4+5     1+2+4+5
                                 1+1+3+5     1+2+2+6     1+3+3+5
                     1+1+1+6     1+1+4+4     1+2+3+5     1+3+4+4
         1+1+1+5     1+1+2+5     1+2+2+5     1+2+4+4     2+2+2+6
         1+1+2+4     1+1+3+4     1+2+3+4     1+3+3+4     2+2+3+5
         1+1+3+3     1+2+2+4     1+3+3+3     2+2+2+5     2+2+4+4
         1+2+2+3     1+2+3+3     2+2+2+4     2+2+3+4     2+3+3+4
         2+2+2+2     2+2+2+3     2+2+3+3     2+3+3+3     3+3+3+3
--------------------------------------------------------------------------
  n  |      8           9          10          11          12        ...
--------------------------------------------------------------------------
a(n) |     10          13          23          30          46        ...
--------------------------------------------------------------------------
- _Wesley Ivan Hurt_, Sep 07 2019

Links

Crossrefs

Cf. A026810, A308733, A308758, A308760, A308775.

Programs

  • Mathematica
    Table[Total[IntegerPartitions[n,{4}][[All,2]]],{n,0,50}] (* Harvey P. Dale, Nov 08 2020 *)

Formula

a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} i.
a(n) = A308775(n) - A308733(n) - A308758(n) - A308760(n).
Conjectures from Colin Barker, Jun 23 2019: (Start)
G.f.: x^4*(1 + x + 2*x^2 + 2*x^3 + 3*x^4 + 2*x^5 + 2*x^6) / ((1 - x)^5*(1 + x)^3*(1 + x^2)^2*(1 + x + x^2)^2).
a(n) = a(n-2) + 2*a(n-3) + 2*a(n-4) - 2*a(n-5) - 3*a(n-6) - 4*a(n-7) + 4*a(n-9) + 3*a(n-10) + 2*a(n-11) - 2*a(n-12) - 2*a(n-13) - a(n-14) + a(n-16) for n>15.
(End)

A308760 Sum of the largest parts of the partitions of n into 4 parts.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 5, 9, 17, 25, 41, 57, 84, 112, 154, 197, 262, 325, 414, 506, 629, 751, 915, 1078, 1289, 1501, 1767, 2034, 2370, 2701, 3108, 3519, 4014, 4506, 5100, 5691, 6393, 7095, 7917, 8739, 9703, 10658, 11765, 12876, 14150, 15418, 16874, 18324, 19974
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 22 2019

Keywords

Examples

			Figure 1: The partitions of n into 4 parts for n = 8, 9, ..
                                                         1+1+1+9
                                                         1+1+2+8
                                                         1+1+3+7
                                                         1+1+4+6
                                             1+1+1+8     1+1+5+5
                                             1+1+2+7     1+2+2+7
                                 1+1+1+7     1+1+3+6     1+2+3+6
                                 1+1+2+6     1+1+4+5     1+2+4+5
                                 1+1+3+5     1+2+2+6     1+3+3+5
                     1+1+1+6     1+1+4+4     1+2+3+5     1+3+4+4
         1+1+1+5     1+1+2+5     1+2+2+5     1+2+4+4     2+2+2+6
         1+1+2+4     1+1+3+4     1+2+3+4     1+3+3+4     2+2+3+5
         1+1+3+3     1+2+2+4     1+3+3+3     2+2+2+5     2+2+4+4
         1+2+2+3     1+2+3+3     2+2+2+4     2+2+3+4     2+3+3+4
         2+2+2+2     2+2+2+3     2+2+3+3     2+3+3+3     3+3+3+3
--------------------------------------------------------------------------
  n  |      8           9          10          11          12        ...
--------------------------------------------------------------------------
a(n) |     17          25          41          57          84        ...
--------------------------------------------------------------------------
- _Wesley Ivan Hurt_, Sep 07 2019

Links

Crossrefs

Cf. A001318, A026810, A308265, A308733, A308758, A308759, A308775.

Programs

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

Formula

a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} (n-i-j-k).
a(n) = A308775(n) - A308733(n) - A308758(n) - A308759(n).
Conjectures from Colin Barker, Jun 23 2019: (Start)
G.f.: x^4*(1 + 2*x + 4*x^2 + 5*x^3 + 6*x^4 + 4*x^5 + 3*x^6) / ((1 - x)^5*(1 + x)^3*(1 + x^2)^2*(1 + x + x^2)^2).
a(n) = a(n-2) + 2*a(n-3) + 2*a(n-4) - 2*a(n-5) - 3*a(n-6) - 4*a(n-7) + 4*a(n-9) + 3*a(n-10) + 2*a(n-11) - 2*a(n-12) - 2*a(n-13) - a(n-14) + a(n-16) for n>15.
(End)

A308775 Sum of all the parts in the partitions of n into 4 parts.

Original entry on oeis.org

0, 0, 0, 0, 4, 5, 12, 21, 40, 54, 90, 121, 180, 234, 322, 405, 544, 663, 846, 1026, 1280, 1512, 1848, 2162, 2592, 3000, 3536, 4050, 4732, 5365, 6180, 6975, 7968, 8910, 10098, 11235, 12636, 13986, 15618, 17199, 19120, 20951, 23142, 25284, 27808, 30240, 33120
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 23 2019

Keywords

Examples

			Figure 1: The partitions of n into 4 parts for n = 8, 9, ..
                                                         1+1+1+9
                                                         1+1+2+8
                                                         1+1+3+7
                                                         1+1+4+6
                                             1+1+1+8     1+1+5+5
                                             1+1+2+7     1+2+2+7
                                 1+1+1+7     1+1+3+6     1+2+3+6
                                 1+1+2+6     1+1+4+5     1+2+4+5
                                 1+1+3+5     1+2+2+6     1+3+3+5
                     1+1+1+6     1+1+4+4     1+2+3+5     1+3+4+4
         1+1+1+5     1+1+2+5     1+2+2+5     1+2+4+4     2+2+2+6
         1+1+2+4     1+1+3+4     1+2+3+4     1+3+3+4     2+2+3+5
         1+1+3+3     1+2+2+4     1+3+3+3     2+2+2+5     2+2+4+4
         1+2+2+3     1+2+3+3     2+2+2+4     2+2+3+4     2+3+3+4
         2+2+2+2     2+2+2+3     2+2+3+3     2+3+3+3     3+3+3+3
--------------------------------------------------------------------------
  n  |      8           9          10          11          12        ...
--------------------------------------------------------------------------
a(n) |     40          54          90         121         180        ...
--------------------------------------------------------------------------
- _Wesley Ivan Hurt_, Sep 07 2019

Links

Crossrefs

Cf. A026810, A308733, A308758, A308759, A308760.

Programs

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

Formula

a(n) = n * A026810(n).
a(n) = A308733(n) + A308758(n) + A308759(n) + A308760(n).
Conjectures from Colin Barker, Jun 24 2019: (Start)
G.f.: x^4*(4 + 5*x + 8*x^2 + 8*x^3 + 10*x^4 + 7*x^5 + 6*x^6) / ((1 - x)^5*(1 + x)^3*(1 + x^2)^2*(1 + x + x^2)^2).
a(n) = a(n-2) + 2*a(n-3) + 2*a(n-4) - 2*a(n-5) - 3*a(n-6) - 4*a(n-7) + 4*a(n-9) + 3*a(n-10) + 2*a(n-11) - 2*a(n-12) - 2*a(n-13) - a(n-14) + a(n-16) for n>15.
(End)
Showing 1-4 of 4 results.

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

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