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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A207377 Triangle read by rows in which row n lists the parts of the last section of the set of partitions of n in nondecreasing order.

Original entry on oeis.org

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

Views

Author

Omar E. Pol, Feb 23 2012

Keywords

Comments

Starting from the first row; it appears that the total numbers of occurrences of k in k successive rows give the sequence A000041. For more information see A182703.

Examples

			Written as a triangle:
1;
1,2;
1,1,3;
1,1,1,2,2,4;
1,1,1,1,1,2,3,5;
1,1,1,1,1,1,1,2,2,2,2,3,3,4,6;
1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,4,5,7;
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,5,6,8;

Links

Crossrefs

Triangle similar to A135010. Mirror of A207378. Row n has length A138137(n). Row sums give A138879. Right border is A000027.
Cf. A000041, A006128, A066186, A138121, A182703, A182712-A182714, A194812.

A210960 Tetrahedron T(j,n,k) in which the slice j is a finite triangle read by rows T(n,k) which list the number of parts in the columns of the shell model of partitions with n shells mentioned in A210970.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 3, 3, 2, 1, 1, 1, 3, 3, 2, 1, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 3, 2, 1, 1, 3, 4, 3, 2, 1, 1, 1, 3, 4, 3, 2, 1, 1
Offset: 1

Views

Author

Omar E. Pol, Apr 22 2012

Keywords

Examples

			--------------------------------------------------------
Illustration of first five
slices of the tetrahedron                       Row sum
--------------------------------------------------------
. 1,                                               1
.    1,                                            1
.    1, 1,                                         2
.          1,                                      1
.          1, 1,                                   2
.          1, 1, 1,                                3
.                   1,                             1
.                   1, 1,                          2
.                   2, 1, 1,                       4
.                   1, 2, 1, 1,                    5
.                               1,                 1
.                               1, 1,              2
.                               2, 1, 1,           4
.                               2, 2, 1, 1,        6
.                               1, 2, 2, 1, 1,     7
--------------------------------------------------------
. 1, 2, 1, 3, 2, 1, 5, 4, 2, 1, 7, 6, 4, 2, 1,
.
It appears that column sums give A058399.
Also, written as a triangle read by rows in which each row is a flattened triangle, begins:
1;
1,1,1,
1,1,1,1,1,1;
1,1,1,2,1,1,1,2,1,1;
1,1,1,2,1,1,2,2,1,1,1,2,2,1,1;
1,1,1,2,1,1,3,2,1,1,3,3,2,1,1,1,3,3,2,1,1;
1,1,1,2,1,1,3,2,1,1,4,3,2,1,1,3,4,3,2,1,1,1,3,4,3,2,1,1;
In which row sums give A006128.

Crossrefs

Cf. A058399, A135010, A138121, A182703, A209655, A209918, A210763, A210961.

A210966 Sum of all region numbers of all parts of the n-th region of the shell model of partitions.

Original entry on oeis.org

1, 4, 9, 4, 25, 6, 49, 8, 18, 10, 121, 12, 26, 14, 225, 16, 34, 18, 76, 20, 21, 484, 23, 48, 25, 104, 27, 56, 29, 900, 31, 64, 33, 136, 35, 36, 259, 38, 78, 40, 41, 1764, 43, 88, 45, 184, 47, 96, 49, 400, 51, 52, 159, 54, 55, 3136, 57, 116, 59, 240
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.

Examples

			The first seven regions of the shell model of partitions (or the seven regions of 5) are [1], [2, 1], [3, 1, 1], [2], [4, 2, 1, 1, 1], [3], [5, 2, 1, 1, 1, 1, 1] therefore the "region numbers" are [1], [2, 2], [3, 3, 3], [4], [5, 5, 5, 5, 5], [6], [7, 7, 7, 7, 7, 7, 7]. So a(1)..a(7) give: 1, 4, 9, 4, 25, 6, 49.
Also written as an irregular triangle the sequence begins:
1;
4;
9;
4,25;
6,49;
8,18,10,121;
12,26,14,225;
16,34,18,76,20,21,484;
23,48,25,104,27,56,29,900;
31,64,33,136,35,36,259,38,78,40,41,1764;
43,88,45,184,47,96,49,400,51,52,159,54,55,3136;

Links

Crossrefs

Row n has length A187219(n). Row sums give A210969. Right border gives A001255, n >= 1.
Cf. A135010, A138121, A194446, A182703, A206437, A210971, A210972.

Formula

a(n) = n*A194446(n).

A211025 Triangle read by rows: T(n,k) = total sum of parts in the last section of the set of partitions of n after k-th stage.

Original entry on oeis.org

1, 1, 3, 1, 2, 5, 1, 2, 3, 5, 7, 11, 1, 2, 3, 4, 5, 7, 10, 15, 1, 2, 3, 4, 5, 6, 7, 9, 11, 13, 15, 19, 22, 25, 31, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 18, 20, 25, 28, 32, 39, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 19, 21, 23, 25, 27
Offset: 1

Views

Author

Omar E. Pol, Apr 25 2012

Keywords

Comments

Also triangle read by rows in which row n lists the partial sums of row n of triangle A135010.
This triangle shows the growth of the last sections of the partitions of n step by step. At stage k one part of size A135010(n,k) is added to the structure of the n-th shell.

Examples

			For row n = 5 of triangle we have:
-------------------------------------
Column  Zone    The 5th    Total sum
k                shell     of parts
-------------------------------------
8    <>   7       (5)         15
7    <>   6    (3...          10
6    =    6     ...2)          7
5    =    5       (1)          5
4    =    4       (1)          4
3    =    3       (1)          3
2    =    2       (1)          2
1    =    1       (1)          1
.
Triangle begins:
1;
1,3;
1,2,5;
1,2,3,5,7,11;
1,2,3,4,5,7,10,15;
1,2,3,4,5,6,7,9,11,13,15,19,22,25,31;
1,2,3,4,5,6,7,8,9,10,11,13,15,18,20,25,28,32,39;

Crossrefs

Row n has length A138137(n). Right border gives A138879.
Cf. A135010, A182703, A206437, A211030.

A211980 Triangle read by rows: T(n,k) = total number of regions in the last n-k+1 shells of n.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 5, 4, 3, 2, 7, 6, 5, 4, 2, 11, 10, 9, 8, 6, 4, 15, 14, 13, 12, 10, 8, 4, 22, 21, 20, 19, 17, 15, 11, 7, 30, 29, 28, 27, 25, 23, 19, 15, 8, 42, 41, 40, 39, 37, 35, 31, 27, 20, 12, 56, 55, 54, 53, 51, 49, 45, 41, 34, 26, 14, 77, 76, 75
Offset: 1

Views

Author

Omar E. Pol, Apr 27 2012

Keywords

Comments

The set of partitions of n contains n shells and A000041(n) regions. For the definition of "last section of n" see A135010. For the definition of "region of n" see A206437.

Examples

			Triangle begins:
1;
2,   1;
3,   2,  1;
5,   4,  3,  2;
7,   6,  5,  4,  2;
11, 10,  9,  8,  6,  4;
15, 14, 13, 12, 10,  8,  4;
22, 21, 20, 19, 17, 15, 11,  7;
30, 29, 28, 27, 25, 23, 19, 15,  8;
42, 41, 40, 39, 37, 35, 31, 27, 20, 12;
56, 55, 54, 53, 51, 49, 45, 41, 34, 26, 14;
77, 76, 75, 74, 72, 70, 66, 62, 55, 47, 35, 21;

Links

Crossrefs

Mirror of triangle A211990. Column 1 is A000041, n >= 1. Right border is A187219.
Cf. A135010, A138121, A182703, A186114, A206437, A212000, A212001.

Formula

T(n,1) = A000041(n).
T(n,k) = A000041(n) - A000041(k-1), 1
T(n,k) = Sum_{j=k..n} A187219(j).

A211986 A list of certain compositions which arise from the ordered partitions of the positive integers in which the shells of each integer are arranged as the arms of a spiral.

Original entry on oeis.org

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

Views

Author

Omar E. Pol, Aug 19 2012

Keywords

Comments

In order to construct this sequence we use the following rules:
- Consider the partitions of positive integers.
- For each positive integer its shells must be arranged as the arms of a spiral.
- The sequence lists one spiral for each positive integer.
- If the integer j is odd then the first composition listed of each spiral is j.
- If the integer j is even then we use the same spiral of A211988.

Examples

			----------------------------------------------
.                 Expanded         Geometric
Compositions     arrangement         model
----------------------------------------------
1;                    1;              |*|
----------------------------------------------
2;                  2 .;            |* *|
1,1;                1,1;            |*|o|
----------------------------------------------
3;                  . . 3;          |* * *|
1,1,1;              1,1,1;          |o|o|*|
2,1;                2 .,1;          |o o|*|
----------------------------------------------
4,;               4 . . .;        |* * * *|
2,2;              2 .,2 .;        |* *|* *|
1,2,1;            1,2 .,1;        |*|o o|o|
1,1,1,1,;         1,1,1,1;        |*|o|o|o|
1,3;              1,. . 3;        |*|o o o|
----------------------------------------------
5;                . . . . 5;      |* * * * *|
3,2;              . . 3,. 2;      |* * *|* *|
1,3,1;            1,. . 3,1;      |o|o o o|*|
1,1,1,1,1;        1,1,1,1,1;      |o|o|o|o|*|
1,2,1,1;          1,2 .,1,1;      |o|o o|o|*|
2,2,1;            2 .,2 .,1;      |o o|o o|*|
4,1;              4 . . .,1;      |o o o o|*|
----------------------------------------------
6;              6 . . . . .;    |* * * * * *|
3,3;            3 . .,3 . .;    |* * *|* * *|
2,4;            2 .,4 . . .;    |* *|* * * *|
2,2,2;          2 .,2 .,2 .;    |* *|* *|* *|
1,4,1;          1,4 . . .,1;    |*|o o o o|o|
1,2,2,1;        1,2 .,2 .,1;    |*|o o|o o|o|
1,1,2,1,1;      1,1,2 .,1,1;    |*|o|o o|o|o|
1,1,1,1,1,1;    1,1,1,1,1,1;    |*|o|o|o|o|o|
1,1,3,1;        1,1,. . 3,1;    |*|o|o o o|o|
1,3,2;          1,. . 3,. 2;    |*|o o o|o o|
1,5;            1,. . . . 5;    |*|o o o o o|
------------------------------------------------
Note that * is a unitary element of every part of the last section of j.

Links

Crossrefs

Rows sums give A036042, n>=1.
Mirror of A211985. Other spiral versions are A211987, A211988, A211995-A211998. See also A026792, A211983, A211984, A211989, A211992, A211993, A211994, A211999.
Cf. A000041, A026905, A135010, A138121, A138137, A138879, A182703, A206437.

A211987 A list of certain compositions which arise from the ordered partitions of the positive integers in which the shells of each integer are arranged as a spiral.

Original entry on oeis.org

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

Views

Author

Omar E. Pol, Aug 18 2012

Keywords

Comments

In order to construct this sequence we use the following rules:
- Consider the partitions of positive integers.
- For each positive integer its shells must be arranged in a spiral.
- The sequence lists one spiral for each positive integer.
- If the integer j is odd then the last composition listed of each spiral is j.
- If the integer j is even then the first composition listed of each spiral is j.
This sequence represents a three-dimensional structure in which each column contains parts of the same size.

Examples

			----------------------------------------------
.                Expanded        Geometric
Compositions    arrangement        model
----------------------------------------------
1;                  1;              |*|
----------------------------------------------
2;                  . 2;            |* *|
1,1;                1,1;            |o|*|
----------------------------------------------
1,2;              1,. 2;          |*|o o|
1,1,1;            1,1,1;          |*|o|o|
3;                3 . .;          |* * *|
----------------------------------------------
4,;               . . . 4;        |* * * *|
2,2;              . 2,. 2;        |* *|* *|
1,2,1;            1,. 2,1;        |o|o o|*|
1,1,1,1,;         1,1,1,1;        |o|o|o|*|
3,1;              3 . .,1;        |o o o|*|
----------------------------------------------
1,4;            1,. . . 4;      |*|o o o o|
1,2,2;          1,. 2,. 2;      |*|o o|o o|
1,1,2,1;        1,1,. 2,1;      |*|o|o o|o|
1,1,1,1,1;      1,1,1,1,1;      |*|o|o|o|o|
1,3,1;          1,3 . .,1;      |*|o o o|o|
2,3;            2 .,3 . .;      |* *|* * *|
5;              5 . . . .;      |* * * * *|
----------------------------------------------
6;              . . . . . 6;    |* * * * * *|
3,3;            . . 3,. . 3;    |* * *|* * *|
4,2;            . . . 4,. 2;    |* * * *|* *|
2,2,2;          . 2,. 2,. 2;    |* *|* *|* *|
1,4,1;          1,. . . 4,1;    |o|o o o o|*|
1,2,2,1;        1,. 2,. 2,1;    |o|o o|o o|*|
1,1,2,1,1;      1,1,. 2,1,1;    |o|o|o o|o|*|
1,1,1,1,1,1;    1,1,1,1,1,1;    |o|o|o|o|o|*|
1,3,1,1;        1,3 . .,1,1;    |o|o o o|o|*|
2,3,1;          2 .,3 . .,1;    |o o|o o o|*|
5,1;            5 . . . .,1;    |o o o o o|*|
----------------------------------------------
Note that * is a unitary element of every part of the last section of j.

Links

Crossrefs

Rows sums give A036042, n>=1.
Mirror of A211988. Other spiral versions are A211985, A211986, A211995-A211998. See also A026792, A211983, A211984, A211989, A211992, A211993, A211994, A211999.
Cf. A000041, A026905, A135010, A138121, A138137, A138879, A182703, A206437.

A228716 Triangle read by rows in which row n lists the rows (including 0's) of the n-th section of the set of partitions (in colexicographic order) of any integer >= n.

Original entry on oeis.org

1, 0, 1, 2, 0, 0, 1, 0, 1, 3, 0, 0, 0, 1, 0, 0, 1, 0, 1, 2, 2, 4, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 3, 2, 5, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 2, 2, 2, 4, 2, 3, 3, 6, 0, 0, 0, 0, 0, 0, 1, 0, 0
Offset: 1

Views

Author

Omar E. Pol, Sep 02 2013

Keywords

Comments

In other words, row n lists the rows of the last section of the set of partitions (in colexicographic order) of n.
Row lengths is A006128.
The number of zeros in row n is A006128(n-1).
Rows sums give A138879.
For more properties of the sections of the set of partitions of a positive integer see example.
Positive terms give A230440. - Omar E. Pol, Oct 25 2013

Examples

			Illustration of the 15 rows of the 7th section (including zeros) of the set of partitions of any integer >= 7 (hence this is also the last section of the set of partitions of 7). Note that the sum of the k-th column is equal to the number of parts >= k, therefore the first differences of the column sums give the number of occurrences of parts k in the section. The same for all sections of all positive integers, see below:
-----------------------------
Column: 1  2  3  4  5  6  7
-----------------------------
Row |
1   |   0, 0, 0, 0, 0, 0, 1;
2   |   0, 0, 0, 0, 0, 1;
3   |   0, 0, 0, 0, 1;
4   |   0, 0, 0, 0, 1;
5   |   0, 0, 0, 1;
6   |   0, 0, 0, 1;
7   |   0, 0, 1;
8   |   0, 0, 0, 1;
9   |   0, 0, 1;
10  |   0, 0, 1;
11  |   0, 1;
12  |   3, 2, 2;
13  |   5, 2;
14  |   4, 3;
15  |   7;
-----------------------------
Sums:  19, 8, 5, 3, 2, 1, 1 -> Row 7 of triangle A207031.
.       | /| /| /| /| /| /|
.       |/ |/ |/ |/ |/ |/ |
F.Dif: 11, 3, 2, 1, 1, 0, 1 -> Row 7 of triangle A182703.
.
Triangle begins:
[1];
[0,1],[2];
[0,0,1],[0,1],[3];
[0,0,0,1],[0,0,1],[0,1],[2,2],[4];
[0,0,0,0,1],[0,0,0,1],[0,0,1],[0,0,1],[0,1],[3,2],[5];
[0,0,0,0,0,1],[0,0,0,0,1],[0,0,0,1],[0,0,0,1],[0,0,1],[0,0,1],[0,1],[2,2,2],[4,2],[3,3],[6];
[0,0,0,0,0,0,1],[0,0,0,0,0,1],[0,0,0,0,1],[0,0,0,0,1],[0,0,0,1],[0,0,0,1],[0,0,1],[0,0,0,1],[0,0,1],[0,0,1],[0,1],[3,2,2],[5,2],[4,3],[7];

Crossrefs

Cf. A000041, A006128, A135010, A138121, A138879, A182703, A187219, A207031, A211992.

A182706 Row sums of triangle A182702.

Original entry on oeis.org

1, 6, 18, 44, 90, 174, 308, 528, 864, 1380, 2134, 3252, 4836, 7098, 10245, 14624, 20587, 28728, 39634, 54260, 73605, 99154, 132526, 176088, 232375, 305006, 398007, 516852, 667696, 858840
Offset: 1

Views

Author

Omar E. Pol, Nov 28 2010

Keywords

Links

Crossrefs

Cf. A026905, A182700, A182701, A182702, A182703, A182704, A182705.

Programs

  • Mathematica
    Total /@ Table[n*PartitionsP[n-k], {n, 30}, {k, 0, n - 1}] // Flatten (* Robert Price, Jun 23 2020 *)

Formula

a(n) = n * A026905(n).

A194450 Vertex number of a rectangular spiral which contains exactly between its edges the successive shells of the partitions of the positive integers.

Original entry on oeis.org

0, 1, 2, 4, 6, 9, 12, 17, 21, 28, 33, 44, 50, 65, 72, 94, 102, 132, 141, 183, 193, 249, 260, 337, 349, 450, 463, 598, 612, 788, 803, 1034, 1050, 1347, 1364, 1749, 1767, 2257, 2276, 2903, 2923, 3715, 3736, 4738, 4760, 6015, 6038, 7613, 7637, 9595
Offset: 0

Views

Author

Omar E. Pol, Nov 01 2011

Keywords

Comments

First differences give A194451, the length of the edges of the spiral. For more information see A135010 and A138121.

Crossrefs

Cf. A000041, A000217, A006128, A026905, A066186, A135010, A138121, A138879, A138880, A182703, A182994, A182995, A194451.

Formula

a(2n-1) = A026905(n) + A000217(n) - n, if n >= 1.
a(2n) = A026905(n) + A000217(n), if n >= 1.
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