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.

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A208476 Triangle read by rows: T(n,k) = total sum of odd/even parts >= k in the last section of the set of partitions of n, if k is odd/even.

Original entry on oeis.org

1, 1, 2, 5, 0, 3, 3, 8, 0, 4, 13, 2, 8, 0, 5, 13, 18, 6, 10, 0, 6
Offset: 1

Views

Author

Omar E. Pol, Feb 28 2012

Keywords

Comments

Essentially this sequence is related to A206562 in the same way as A207032 is related to A207031 and also in the same way as A206563 is related to A181187. See the calculation in the example section of A206563.

Examples

			Triangle begins:
1;
1,   2;
5,   0,  3;
3,   8,  0,  4;
13,  2,  8,  0,  5;
13, 18,  6, 10,  0,  6;

Crossrefs

Right border is A000027.
Cf. A135010, A182703, A206562, A207031, A207032, A207383, A208475.

A210955 Triangle read by rows: T(n,k) = total number of parts <= k in the last section of the set of partitions of n.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 3, 5, 5, 6, 5, 6, 7, 7, 8, 7, 11, 13, 14, 14, 15, 11, 14, 16, 17, 18, 18, 19, 15, 23, 26, 29, 30, 31, 31, 32, 22, 29, 35, 37, 39, 40, 41, 41, 42, 30, 45, 51, 56, 59, 61, 62, 63, 63, 64, 42, 57, 67, 72, 76, 78, 80, 81, 82, 82, 83
Offset: 1

Views

Author

Omar E. Pol, May 01 2012

Keywords

Comments

Row n lists the partial sums of row n of triangle A182703.

Examples

			1,
1,   2,
2,   2,  3,
3,   5,  5,  6,
5,   6,  7,  7,  8,
7,  11, 13, 14, 14, 15,
11, 14, 16, 17, 18, 18, 19,
15, 23, 26, 29, 30, 31, 31, 32,
22, 29, 35, 37, 39, 40, 41, 41, 42;

Crossrefs

Column 1 is A000041. Right border gives A138137.
Cf. A135010, A138121, A182703, A206437, A206562, A207031, A207032, A207383, A208476, A210947, A210956.

Formula

T(n,k) = Sum_{j=1..k} A182703(n,j).

Extensions

More terms from Alois P. Heinz, May 25 2013

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.

A210971 Triangle read by rows in which row n lists the region number of the parts of the k-th partition of n, with partitions reverse lexicographically ordered.

Original entry on oeis.org

1, 3, 2, 6, 5, 3, 11, 10, 8, 9, 5, 18, 17, 15, 16, 12, 13, 7, 29, 28, 26, 27, 23, 24, 18, 28, 20, 21, 11
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             Sum of
k           of the partitions of 5        partition k
------------------------------------------------------
7      [5]                          [5]        5
6      [3+2]                      [3+2]        5
5      [4+1]                    [4  +1]        5
4      [2+1+1]                [2+2  +1]        5
3      [3+1+1]              [3  +1  +1]        5
2      [2+1+1+1]          [2+1  +1  +1]        5
1      [1+1+1+1+1]      [1+1+1  +1  +1]        5
------------------------------------------------------
.              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
------------------------------------------------------
So row 5 of triangle gives: 18, 17, 15, 16, 12, 13, 7.
.
Triangle begins:
1;
3,2;
6,5,3;
11,10,8,9,5;
18,17,15,16,12,13,7;
29,28,26,27,23,24,18,28,20,21,11;

Links

Crossrefs

Column 1 is A026905. Right border = row lengths = A000041, n>=1. Row sums give A210972.
Cf. A135010, A138121, A182703, A194446, A206437, A210969.

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.

A230440 Triangle read by rows in which row n lists A000041(n-1) 1's followed by the list of partitions of n that do not contain 1 as a part in colexicographic order.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 1, 1, 2, 2, 4, 1, 1, 1, 1, 1, 3, 2, 5, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 4, 2, 3, 3, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 2, 2, 5, 2, 4, 3, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 4, 2, 2, 3, 3, 2, 6, 2, 5, 3, 4, 4, 8
Offset: 1

Views

Author

Omar E. Pol, Oct 18 2013

Keywords

Comments

The n-th row of triangle lists the parts of the n-th section of the set of partitions of any integer >= n. For the definition of "section" see A135010.

Examples

			Illustration of initial terms (row = 1..6). The table shows the six sections of the set of partitions of 6 in three ways. Note that before the dissection, the set of partitions was in colexicographic order, see A211992. More generally, in a master model, the six sections of the set of partitions of 6 also can be interpreted as the first six sections of the set of partitions of any integer >= 6.
---------------------------------------------------------
n  j     Diagram          Parts              Parts
---------------------------------------------------------
.         _
1  1     |_|              1;                 1;
.           _
2  1      _| |              1,                 1,
2  2     |_ _|              2;               2;
.             _
3  1         | |              1,                 1,
3  2      _ _| |              1,               1,
3  3     |_ _ _|              3;             3;
.               _
4  1           | |              1,                 1,
4  2           | |              1,               1,
4  3      _ _ _| |              1,             1,
4  4     |_ _|   |            2,2,           2,2,
4  5     |_ _ _ _|              4;           4;
.                 _
5  1             | |              1,                 1,
5  2             | |              1,               1,
5  3             | |              1,             1,
5  4             | |              1,             1,
5  5      _ _ _ _| |              1,           1,
5  6     |_ _ _|   |            3,2,         3,2,
5  7     |_ _ _ _ _|              5;         5;
.                   _
6  1               | |              1,                 1,
6  2               | |              1,               1,
6  3               | |              1,             1,
6  4               | |              1,             1,
6  5               | |              1,           1,
6  6               | |              1,           1,
6  7      _ _ _ _ _| |              1,         1,
6  8     |_ _|   |   |          2,2,2,       2,2,2,
6  9     |_ _ _ _|   |            4,2,       4,2,
6  10    |_ _ _|     |            3,3,       3,3,
6  11    |_ _ _ _ _ _|              6;       6;
...
Triangle begins:
[1];
[1],[2];
[1],[1],[3];
[1],[1],[1],[2,2],[4];
[1],[1],[1],[1],[1],[3,2],[5];
[1],[1],[1],[1],[1],[1],[1],[2,2,2],[4,2],[3,3],[6];
...

Crossrefs

Positive terms of A228716.
Row n has length A138137(n).
Row sums give A138879.
Right border gives A000027.
Cf. A000041, A135010, A138121, A141285, A182703, A187219, A193870, A194446, A206437, A207031, A207034, A207383, A207379, A211009.

A182722 a(n) = A005291(n+1)-A182712(n).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 5, 10, 14, 26, 36, 60, 83, 128, 175, 261, 351, 504, 674, 943, 1247, 1711, 2243
Offset: 0

Views

Author

Omar E. Pol, Nov 28 2010, Nov 29 2010

Keywords

Comments

The difference between two apparently unrelated sequences which happen to have the same initial terms. - N. J. A. Sloane, Dec 01 2010

Crossrefs

Cf. A005291, A087787, A100818, A135010, A138121, A182703, A182712.

Formula

a(n) = A005291(n+1)-A182712(n)

A194703 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (3 + m).

Original entry on oeis.org

3, 2, 1, 0, 1, 2, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 3, 2, 1, 0, 1, 2, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Omar E. Pol, Feb 05 2012

Keywords

Comments

Sub-triangle of A182703 and also of A194812. Note that the sum of every row is also the number of partitions of 3. For further information see A182703 and A135010.

Examples

			Triangle begins:
3,
2, 1,
0, 1, 2,
1, 0, 1, 1,
0, 1, 0, 1, 1,
0, 0, 1, 0, 1, 1,
0, 0, 0, 1, 0, 1, 1,
0, 0, 0, 0, 1, 0, 1, 1,
0, 0, 0, 0, 0, 1, 0, 1, 1,
0, 0, 0, 0, 0, 0, 1, 0, 1, 1,
...
For k = 1 and m = 1, T(1,1) = 3 because there are three parts of size 1 in the last section of the set of partitions of 4, since 3 + m = 4, so a(1) = 3.
For k = 2 and m = 1, T(2,1) = 2 because there are two parts of size 2 in the last section of the set of partitions of 4, since 3 + m = 4, so a(2) = 2.

Crossrefs

Always the sum of row k = p(3) = A000041(3) = 3.
The first (0-10) members of this family of triangles are A023531, A129186, A194702, this sequence, A194704-A194710.
Cf. A135010, A138121, A182712-A182714, A194812.

Formula

T(k,m) = A182703(3+m,k), with T(k,m) = 0 if k > 3+m.
T(k,m) = A194812(3+m,k).

A206556 Number of 6's in the last section of the set of partitions of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 5, 4, 8, 9, 14, 16, 26, 28, 42, 50, 69, 82, 114, 133, 179, 215, 279, 335, 434, 516, 657, 789, 987, 1182, 1473, 1754, 2164, 2583, 3154, 3755, 4567, 5414, 6542, 7753, 9307, 11000, 13158, 15501, 18456, 21712, 25731, 30196, 35677
Offset: 1

Views

Author

Omar E. Pol, Feb 09 2012

Keywords

Comments

Zero together with the first differences of A024790. Also number of occurrences of 6 in all partitions of n that do not contain 1 as a part. For the definition of "last section of n" see A135010. It appears that the sums of six successive terms give the partition numbers A000041.

Crossrefs

Column 6 of A182703 and of A194812.
Cf. A000041, A135010, A138121, A182712-A182714, A206556-A206560.

Programs

  • Sage
    A206556 = lambda n: sum(list(p).count(6) for p in Partitions(n) if 1 not in p)

Formula

It appears that A000041(n) = Sum_{j=1..6} a(n+j), n >= 0.

A206557 Number of 7's in the last section of the set of partitions of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 5, 7, 9, 13, 16, 23, 28, 39, 48, 64, 79, 104, 128, 165, 204, 258, 317, 399, 487, 606, 739, 912, 1105, 1356, 1637, 1994, 2400, 2906, 3485, 4199, 5016, 6015, 7164, 8553, 10151, 12076, 14286, 16930, 19974, 23588, 27749
Offset: 1

Views

Author

Omar E. Pol, Feb 09 2012

Keywords

Comments

Zero together with the first differences of A024791. Also number of occurrences of 7 in all partitions of n that do not contain 1 as a part. For the definition of "last section of n" see A135010. It appears that the sums of seven successive terms give the partition numbers A000041.

Crossrefs

Column 7 of A182703 and of A194812.
Cf. A000041, A135010, A138121, A182712-A182714, A206555, A206556, A206558-A206560.

Programs

  • Sage
    A206557 = lambda n: sum(list(p).count(7) for p in Partitions(n) if 1 not in p)

Formula

It appears that A000041(n) = Sum_{j=1..7} a(n+j), n >= 0.
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