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 1-2 of 2 results.

A210969 Sum of all region numbers of all parts of the last section of the set of partitions of n.

Original entry on oeis.org

1, 4, 9, 29, 55, 157, 277, 669, 1212, 2555, 4459, 9048
Offset: 1

Views

Author

Omar E. Pol, Jul 01 2012

Keywords

Comments

Each part of a partition of n belongs to a different region of n. The "region number" of a part of the r-th region of n is equal to r. For the definition of "region of n" see A206437.
The last section of the set of partitions of n is also the n-th section of the set of partitions of any integer >= n. - Omar E. Pol, Apr 07 2014

Examples

			For n = 6 the four regions of the last section of 6 are [2], [4, 2], [3], [6, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1] therefore the "region numbers" are [8], [9, 9], [10], [11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11]. The sum of all region numbers is a(6) = 8+2*9+10+11^2 =  8+18+10+121 = 157, see below:
--------------------------------------------
.     Last section                  Sum of
.     of the set of     Region      region
k    partitions of 6    numbers     numbers
--------------------------------------------
11           6              11         11
10         3+3           10,11         21
9        4  +2         9,   11         20
8      2+2  +2       8,9,   11         28
7            1              11         11
6            1              11         11
5            1              11         11
4            1              11         11
3            1              11         11
2            1              11         11
1            1              11         11
--------------------------------------------
Total sum of region numbers is a(6) = 157

Links

Crossrefs

Row sums of triangle A210966. Partial sums give A210972.
Cf. A135010, A138121, A194446, A182703, A206437, A210971.

A210972 Sum of all region numbers of all parts of all partitions of n.

Original entry on oeis.org

1, 5, 14, 43, 98, 255, 532, 1201, 2413, 4968, 9427, 18475
Offset: 1

Views

Author

Omar E. Pol, Jun 30 2012

Keywords

Comments

Each part of a partition of n belongs to a different region of n. The "region number" of a part of the r-th region of n is equal to r. For the definition of "region of n" see A206437.

Examples

			For n = 5 we have:
---------------------------------------------------
.              Two arrangements
k           of the partitions of 5
---------------------------------------------------
7      [5]                          [5]
6      [3+2]                      [3+2]
5      [4+1]                    [4  +1]
4      [2+1+1]                [2+2  +1]
3      [3+1+1]              [3  +1  +1]
2      [2+1+1+1]          [2+1  +1  +1]
1      [1+1+1+1+1]      [1+1+1  +1  +1]
---------------------------------------------------
.              Two arrangements
.           of the region numbers           Sum of
k           of the partitions of 5          zone k
---------------------------------------------------
7      [7]                          [7]        7
6      [6,7]                      [6,7]       13
5      [5,7]                    [5,  7]       12
4      [4,5,7]                [4,5,  7]       16
3      [3,5,7]              [3,  5,  7]       15
2      [2,3,5,7]          [2,3,  5,  7]       17
1      [1,2,3,5,7]      [1,2,3,  5,  7]       18
---------------------------------------------------
The total sum is a(5) = 1+2^2+3^2+4+5^2+6+7^2 = 1+4+9+4+25+6+49 = 18+17+15+16+12+13+7 = 98.

Links

Crossrefs

Partial sums of A210969. Row sums of triangle A210971.
Cf. A135010, A138121, A182703, A194446, A210437, A210966.
Showing 1-2 of 2 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.