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 1-6 of 6 results.

A308867 Sum of all the parts in the partitions of n into 6 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 6, 7, 16, 27, 50, 77, 132, 182, 280, 390, 560, 748, 1044, 1349, 1800, 2310, 2992, 3749, 4776, 5875, 7332, 8937, 10948, 13166, 15960, 18972, 22688, 26763, 31654, 36995, 43416, 50320, 58520, 67431, 77800, 89052, 102144, 116186, 132396, 149895
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 29 2019

Keywords

Links

Crossrefs

Cf. A238340, A308868, A308869, A306670, A306671, A308872, A308873.

Programs

  • Mathematica
    Table[n*Sum[Sum[Sum[Sum[Sum[1, {i, j, Floor[(n - j - k - l - m)/2]}], {j, k, Floor[(n - k - l - m)/3]}], {k, l, Floor[(n - l - m)/4]}], {l, m, Floor[(n - m)/5]}], {m, Floor[n/6]}], {n, 0, 100}]
Table[Total[Flatten[IntegerPartitions[n,{6}]]],{n,0,50}] (* Harvey P. Dale, Oct 29 2024 *)

Formula

a(n) = n * Sum_{m=1..floor(n/6)} Sum_{l=m..floor((n-m)/5)} Sum_{k=l..floor((n-l-m)/4)} Sum_{j=k..floor((n-k-l-m)/3)} Sum_{i=j..floor((n-j-k-l-m)/2)} 1.
a(n) = n * A238340(n).
a(n) = A308868(n) + A308869(n) + A306670(n) + A306671(n) + A308872(n) + A308873(n).

A308868 Sum of the smallest parts in the partitions of n into 6 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 12, 15, 22, 29, 40, 51, 70, 86, 112, 139, 176, 214, 269, 321, 394, 470, 567, 668, 801, 933, 1103, 1281, 1498, 1725, 2007, 2293, 2643, 3010, 3443, 3897, 4439, 4995, 5652, 6341, 7135, 7967, 8933, 9930, 11079, 12283, 13645
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 29 2019

Keywords

Links

Crossrefs

Cf. A238340, A308867, A308869, A306670, A306671, A308872, A308873.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[m, {i, j, Floor[(n - j - k - l - m)/2]}], {j, k, Floor[(n - k - l - m)/3]}], {k, l, Floor[(n - l - m)/4]}], {l, m, Floor[(n - m)/5]}], {m, Floor[n/6]}], {n, 0, 50}]

Formula

a(n) = n * Sum_{m=1..floor(n/6)} Sum_{l=m..floor((n-m)/5)} Sum_{k=l..floor((n-l-m)/4)} Sum_{j=k..floor((n-k-l-m)/3)} Sum_{i=j..floor((n-j-k-l-m)/2)} m.
a(n) = A308867(n) - A308869(n) - A306670(n) - A306671(n) - A308872(n) - A308873(n).

A308869 Sum of the fifth largest parts in the partitions of n into 6 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 8, 13, 17, 25, 34, 48, 63, 86, 109, 143, 182, 232, 288, 363, 442, 547, 662, 804, 961, 1157, 1368, 1626, 1909, 2245, 2613, 3054, 3525, 4082, 4688, 5388, 6150, 7031, 7974, 9059, 10231, 11560, 12991, 14614, 16346, 18300, 20400
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 29 2019

Keywords

Links

Crossrefs

Cf. A238340, A308867, A308868, A306670, A306671, A308872, A308873.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[l, {i, j, Floor[(n - j - k - l - m)/2]}], {j, k, Floor[(n - k - l - m)/3]}], {k, l, Floor[(n - l - m)/4]}], {l, m, Floor[(n - m)/5]}], {m, Floor[n/6]}], {n, 0, 50}]

Formula

a(n) = Sum_{m=1..floor(n/6)} Sum_{l=m..floor((n-m)/5)} Sum_{k=l..floor((n-l-m)/4)} Sum_{j=k..floor((n-k-l-m)/3)} Sum_{i=j..floor((n-j-k-l-m)/2)} l.
a(n) = A308867(n) - A308868(n) - A306670(n) - A306671(n) - A308872(n) - A308873(n).

A308872 Sum of the second largest parts in the partitions of n into 6 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 3, 5, 10, 15, 27, 37, 59, 82, 120, 160, 227, 293, 396, 508, 664, 832, 1068, 1314, 1650, 2012, 2477, 2980, 3628, 4314, 5178, 6111, 7250, 8477, 9975, 11566, 13483, 15543, 17970, 20577, 23646, 26907, 30712, 34785, 39469, 44472, 50217
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 29 2019

Keywords

Links

Crossrefs

Cf. A238340, A308867, A308868, A308869, A306670, A306671, A308873.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[i, {i, j, Floor[(n - j - k - l - m)/2]}], {j, k, Floor[(n - k - l - m)/3]}], {k, l, Floor[(n - l - m)/4]}], {l, m, Floor[(n - m)/5]}], {m, Floor[n/6]}], {n, 0, 50}]

Formula

a(n) = Sum_{m=1..floor(n/6)} Sum_{l=m..floor((n-m)/5)} Sum_{k=l..floor((n-l-m)/4)} Sum_{j=k..floor((n-k-l-m)/3)} Sum_{i=j..floor((n-j-k-l-m)/2)} i.
a(n) = A308867(n) - A308868(n) - A308869(n) - A306670(n) - A306671(n) - A308873(n).

A308870 Sum of the fourth largest parts in the partitions of n into 6 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 6, 9, 15, 20, 31, 42, 61, 80, 112, 143, 191, 243, 316, 393, 501, 613, 767, 930, 1141, 1367, 1659, 1967, 2354, 2769, 3279, 3824, 4491, 5196, 6047, 6956, 8031, 9181, 10536, 11971, 13647, 15434, 17497, 19690, 22211, 24880, 27929
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 29 2019

Keywords

Links

Crossrefs

Cf. A238340, A308867, A308868, A308869, A306671, A308872, A308873.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[k, {i, j, Floor[(n - j - k - l - m)/2]}], {j, k, Floor[(n - k - l - m)/3]}], {k, l, Floor[(n - l - m)/4]}], {l, m, Floor[(n - m)/5]}], {m, Floor[n/6]}], {n, 0, 50}]
Table[Total[IntegerPartitions[n,{6}][[;;,4]]],{n,0,50}] (* Harvey P. Dale, Jul 30 2024 *)

Formula

a(n) = Sum_{m=1..floor(n/6)} Sum_{l=m..floor((n-m)/5)} Sum_{k=l..floor((n-l-m)/4)} Sum_{j=k..floor((n-k-l-m)/3)} Sum_{i=j..floor((n-j-k-l-m)/2)} k.
a(n) = A308867(n) - A308868(n) - A308869(n) - A306671(n) - A308872(n) - A308873(n).

A308871 Sum of the third largest parts in the partitions of n into 6 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 7, 11, 19, 26, 40, 57, 81, 109, 153, 198, 264, 342, 442, 556, 710, 875, 1093, 1338, 1638, 1975, 2398, 2855, 3416, 4040, 4779, 5595, 6573, 7627, 8875, 10244, 11822, 13549, 15553, 17707, 20187, 22883, 25935, 29239, 32991, 37010
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 29 2019

Keywords

Links

Crossrefs

Cf. A238340, A308867, A308868, A308869, A306670, A308872, A308873.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[j, {i, j, Floor[(n - j - k - l - m)/2]}], {j, k, Floor[(n - k - l - m)/3]}], {k, l, Floor[(n - l - m)/4]}], {l, m, Floor[(n - m)/5]}], {m, Floor[n/6]}], {n, 0, 50}]

Formula

a(n) = Sum_{m=1..floor(n/6)} Sum_{l=m..floor((n-m)/5)} Sum_{k=l..floor((n-l-m)/4)} Sum_{j=k..floor((n-k-l-m)/3)} Sum_{i=j..floor((n-j-k-l-m)/2)} j.
a(n) = A308867(n) - A308868(n) - A308869(n) - A306670(n) - A308872(n) - A308873(n).
Showing 1-6 of 6 results.

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