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 1-9 of 9 results.

A209655 Tetrahedron in which the n-th slice is also one of the three views of the shell model of partitions of A207380 with n shells.

Original entry on oeis.org

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

Views

Author

Omar E. Pol, Mar 25 2012

Keywords

Comments

Each slice of the tetrahedron is a triangle, thus the number of elements in the n-th slice is A000217(n). The slices are perpendicular to the slices of A026792. Each element of the n-th slice equals the volume of a column of the shell model of partitions with n shells. The sum of each row of the n-th slice is A000041(n). The sum of all elements of the n-th slice is A066186(n).
It appears that the triangle formed by the last row of each slice gives A008284 and A058398.
It appears that the triangle formed by the first column of each slice gives A058399.
Also consider a vertical rectangle on the infinite square grid with shorter side = n and longer side = p(n) = A000041(n). Each row of rectangle represents a partition of n. Each part of each partition of n is a horizontal rectangle with shorter side = 1 and longer side = k, where k is the size of the part. It appears that T(n,k,j) is also the number of k-th parts of all partitions of n in the j-th column of rectangle.

Examples

			--------------------------------------------------------
Illustration of first five
slices of the tetrahedron                       Row sum
--------------------------------------------------------
. 1,                                               1
.    2,                                            2
.    1, 1,                                         2
.          3,                                      3
.          2, 1,                                   3
.          1, 1, 1,                                3
.                   5,                             5
.                   4, 1,                          5
.                   2, 2, 1,                       5
.                   1, 2, 1, 1,                    5
.                               7,                 7
.                               6, 1,              7
.                               4, 2, 1,           7
.                               2, 3, 1, 1,        7
.                               1, 2, 2, 1, 1,     7
--------------------------------------------------------
. 1, 3, 1, 6, 2, 1,12, 5, 2, 1,20, 8, 4, 2, 1,
.
Written as a triangle begins:
1;
2, 1, 1;
3, 2, 1, 1, 1, 1;
5, 4, 1, 2, 2, 1, 1, 2, 1, 1;
7, 6, 1, 4, 2, 1, 2, 3, 1, 1, 1, 2, 2, 1, 1;
In which row sums give A066186.

Crossrefs

Column sums give A181187. Main diagonal gives A210765. Another version is A209918.
Cf. A000041, A000217, A002260, A004736, A008284, A026792, A058398, A058399, A066186, A135010, A182703, A182715, A207380.

A209918 Tetrahedron in which the n-th slice is also one of the three views of the shell model of partitions of A207380 with n shells.

Original entry on oeis.org

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

Views

Author

Omar E. Pol, Mar 26 2012

Keywords

Comments

Each slice of the tetrahedron is a triangle, thus the number of elements in the n-th slice is A000217(n). The slices are perpendicular to the slices of A026792. Each element of the n-th slice equals the volume of a column of the shell model of partitions with n shells. The sum of each column of the n-th slice is A000041(n). The sum of all elements of the n-th slice is A066186(n).
It appears that the triangle formed by the first row of each slice gives A058399.
It appears that the triangle formed by the last column of each slice gives A008284 and A058398.
Also consider a vertical rectangle on the infinite square grid with shorter side = n and longer side = p(n) = A000041(n). Each row of rectangle represents a partition of n. Each part of each partition of n is a horizontal rectangle with shorter side = 1 and longer side = k, where k is the size of the part. It appears that T(n,k,j) is also the number of k-th parts of all partitions of n in the j-th column of rectangle.

Examples

			---------------------------------------------------------
Illustration of first five                       A181187
slices of the tetrahedron                        Row sum
---------------------------------------------------------
. 1,                                                1
.    2, 1,                                          3
.       1,                                          1
.          3, 2, 1                                  6
.             1, 1,                                 2
.                1,                                 1
.                   5, 4, 2, 1,                    12
.                      1, 2, 2,                     5
.                         1, 1                      2
.                            1,                     1
.                               7, 6, 4, 2, 1,     20
.                                  1, 2, 3, 2,      8
.                                     1, 1, 2,      4
.                                        1, 1,      2
.                                           1,      1
--------------------------------------------------------
. 1, 2, 2, 3, 3, 3, 5, 5, 5, 5, 7, 7, 7, 7, 7,
.
Note that the 5th slice appears as one of three views of the model in the example section of A207380.

Crossrefs

Row sums give A181187. Column sums give A209656. Main diagonal gives A210765. Another version is A209655.
Cf. A000041, A000217, A002260, A004736, A008284, A026792, A058398, A058399, A066186, A135010, A182703, A182715, A207380.

A210970 Total area of the shadows of the three views of a three-dimensional version of the shell model of partitions with n shells.

Original entry on oeis.org

0, 3, 9, 18, 34, 55, 91, 136, 208, 301, 439, 616, 876, 1203, 1665, 2256, 3062, 4083, 5459, 7186, 9470, 12335, 16051, 20688, 26648, 34027, 43395, 54966, 69496, 87341, 109591, 136766, 170382, 211293, 261519, 322382, 396694, 486327, 595143, 725954, 883912
Offset: 0

Views

Author

Omar E. Pol, Apr 22 2012

Keywords

Comments

For more information see A135010 and A182703.

Examples

			For n = 6 the illustration of the three views of a three-dimensional version of the shell model of partitions with 6 shells looks like this:
.
.   A006128(6) = 35     A006128(6) = 35
.
.                 6     6
.               3 3     3 3
.               4 2     4 2
.             2 2 2     2 2 2
.               5 1     5 1
.             3 2 1     3 2 1
.             4 1 1     4 1 1
.           2 2 1 1     2 2 1 1
.           3 1 1 1     3 1 1 1
.         2 1 1 1 1     2 1 1 1 1
.       1 1 1 1 1 1     1 1 1 1 1 1
.
.
.       1 2 5 9 12 6  \
.         1 1 3 5 6    \
.           1 1 2 4     \ 6th slice of
.             1 1 2     / tetrahedron A210961
.               1 1    /
.                 1   /
.
.      A000217(6) = 21
.
The areas of the shadows of the three views are A006128(6) = 35, A006128(6) = 35 and A000217(6) = 21, therefore the total area of the three shadows is 35+35+21 = 91, so a(6) = 91.

Crossrefs

Other versions: A207380, A210979, A210980, A210990, A210991.
Cf. A000217, A006128, A066186, A135010, A138121, A182703, A206437, A210952, A210961.

Formula

a(n) = 2*A006128(n) + A000217(n).

A210990 Total area of the shadows of the three views of the shell model of partitions with n regions.

Original entry on oeis.org

0, 3, 10, 21, 26, 44, 51, 75, 80, 92, 99, 136, 143, 157, 166, 213, 218, 230, 237, 260, 271, 280, 348, 355, 369, 378, 403, 410, 427, 438, 526, 531, 543, 550, 573, 584, 593, 631, 640, 659, 672, 683, 804, 811, 825, 834, 859, 866, 883, 894, 938, 949, 958
Offset: 0

Views

Author

Omar E. Pol, Apr 23 2012

Keywords

Comments

Each part is represented by a cuboid of sides 1 X 1 X k where k is the size of the part. For the definition of "regions of n" see A206437.

Examples

			For n = 11 the three views of the shell model of partitions with 11 regions look like this:
.
.   A182181(11) = 35            A182244(11) = 66
.
.   6                             * * * * * 6
.   3 3                      P    * * 3 * * 3
.   2   4                    a    * * * 4 * 2
.   2   2 2                  r    * 2 * 2 * 2
.   1       5                t    * * * * 5 1
.   1       2 3              i    * * 3 * 2 1
.   1       1   4            t    * * * 4 1 1
.   1       1   2 2          i    * 2 * 2 1 1
.   1       1   1   3        o    * * 3 1 1 1
.   1       1   1   1 2      n    * 2 1 1 1 1
.   1       1   1   1 1 1    s    1 1 1 1 1 1
. <------- Regions ------         ------------> N
.                            L
.                            a    1
.                            r    * 2
.                            g    * * 3
.                            e    * 2
.                            s    * * * 4
.                            t    * * 3
.                                 * * * * 5
.                            p    * 2
.                            a    * * * 4
.                            r    * * 3
.                            t    * * * * * 6
.                            s
.                               A182727(11) = 35
.
The areas of the shadows of the three views are A182244(11) = 66, A182181(11) = 35 and A182727(11) = 35, therefore the total area of the three shadows is 66+35+35 = 136, so a(11) = 136.

Crossrefs

Other versions: A207380, A210970, A210979, A210980, A210991.
Cf. A135010, A138121, A141285, A194446, A182181, A182727, A186114, A206437.

Formula

a(n) = A182244(n) + A182727(n) + A182181(n), n >= 1.
a(A000041(n)) = 2*A006128(n) + A066186(n).

A210979 Total area of the shadows of the three views of the version "Tree" of the shell model of partitions with n shells.

Original entry on oeis.org

0, 3, 8, 15, 27, 42, 69, 102, 155, 225, 327, 458, 652, 894, 1232, 1669, 2257, 2999, 3996, 5242, 6877, 8928, 11564, 14845, 19045, 24223, 30756, 38815, 48877, 61195, 76496, 95124, 118067, 145930, 179991, 221160, 271268, 331538, 404463, 491948, 597253
Offset: 0

Views

Author

Omar E. Pol, Apr 28 2012

Keywords

Comments

The physical model shows each part of a partition as an object, for example; a cube of side 1 which is labeled with the size of the part. Note that on the branches of the tree each column contains parts of the same size, as a periodic structure. For the large version of this model see A210980.

Examples

			For n = 7 the three views of the shell model of partitions version "tree" with seven shells looks like this:
.
.         A194805(7) = 25        A006128(7) = 54
.
.                        7       7
.                      4         4 3
.                    5           5 2
.                  3             3 2 2
.        6       1               6 1
.          3     1               3 3 1
.            4   1               4 2 1
.              2 1               2 2 2 1
.                1   5           5 1 1
.                1 3             3 2 1 1
.            4   1               4 1 1 1
.              2 1               2 2 1 1 1
.                1 3             3 1 1 1 1
.              2 1               2 1 1 1 1 1
.                1               1 1 1 1 1 1 1
-------------------------------------------------
.
.        6 3 4 2 1 3 5 4 7
.          3 2 2 1 2 2 3
.              2 1 2
.                1
.                1
.                1
.                1
.
.         A194803(7) = 23
.
The areas of the shadows of the three views are A006128(7) = 54, A194803(7) = 23 and A194805(7) = 25, therefore the total area of the three shadows is 54+23+25 = 102, so a(7) = 102.

Crossrefs

Other versions: A207380, A210970, A210980, A210990, A210991.
Cf. A006128, A135010, A138121, A182703, A194803, A194805.

Formula

a(n) = A006128(n) + A194803(n) + A194805(n).

A210980 Total area of the shadows of the three views of the shell model of partitions, version "Tree", with n shells.

Original entry on oeis.org

0, 3, 10, 21, 42, 69, 123, 189, 304, 458, 693, 998, 1474, 2067, 2927, 4056, 5613, 7595, 10335, 13782, 18411, 24276, 31944, 41583, 54152, 69762, 89758, 114668, 146181, 185083, 234051, 294126, 368992, 460669, 573906, 711865, 881506, 1087023, 1338043
Offset: 0

Views

Author

Omar E. Pol, Apr 21 2012

Keywords

Comments

Each part is represented by a cuboid 1 X 1 X L where L is the size of the part.

Examples

			For n = 7 the shadows of the three views of the shell model of partitions version "tree" with seven shells looks like this:
.                                        |  Partitions
.    A194805(7) = 25    A066186(7) = 105 |  of 7
.                                        |
.                   1    * * * * * * 1   |  7
.                 2      * * * 1 * * 2   |  4+3
.               2        * * * * 1 * 2   |  5+2
.             3          * * 1 * 2 * 3   |  3+2+2
.   1       2            * * * * * 1 2   |  6+1
.     2     3            * * 1 * * 2 3   |  3+3+1
.       2   3            * * * 1 * 2 3   |  4+2+1
.         3 4            * 1 * 2 * 3 4   |  2+2+2+1
.           3   1        * * * * 1 2 3   |  5+1+1
.           4 2          * * 1 * 2 3 4   |  3+2+1+1
.       1   4            * * * 1 2 3 4   |  4+1+1+1
.         2 5            * 1 * 2 3 4 5   |  2+2+1+1+1
.           5 1          * * 1 2 3 4 5   |  3+1+1+1+1
.         1 6            * 1 2 3 4 5 6   |  2+1+1+1+1+1
.           7            1 2 3 4 5 6 7   |  1+1+1+1+1+1+1
.   ----------------------------------   |
.                                        |
.   * * * * 1 * * * *                    |
.   * * * 1 2 * * * *                    |
.   * 1 * * 2 1 * * *                    |
.   * * 1 2 2 * * 1 *                    |
.   * * * * 2 2 1 * *                    |
.   1 2 2 3 2 * * * *                    |
.           2 3 2 2 1                    |
.                                        |
.    A194804(7) = 59                     |
.
Note that, as a variant, in this case each part is labeled with its position in the partition.
The areas of the shadows of the three views are A066186(7) = 105, A194804(7) = 59 and A194805(7) = 25, therefore the total area of the three shadows is 105+59+25 = 189, so a(7) = 189.

Crossrefs

Other versions: A207380, A210970, A210979, A210990, A210991.
Cf. A000041, A066186, A135010, A138121, A141285, A182703, A194804, A194805, A206437.

Formula

a(n) = A066186(n) + A194804(n) + A194805(n), n >= 1.

A210991 Total area of the shadows of the three views of the shell model of partitions with n regions.

Original entry on oeis.org

0, 3, 9, 18, 21, 35, 39, 58, 61, 67, 71, 99, 103, 110, 115, 152, 155, 161, 165, 175, 181, 186, 238, 242, 249, 254, 265, 269, 277, 283, 352, 355, 361, 365, 375, 381, 386, 401, 406, 415, 422, 428, 522, 526, 533, 538, 549, 553, 561, 567, 584, 590, 595, 606
Offset: 0

Views

Author

Omar E. Pol, Apr 30 2012

Keywords

Comments

It appears that if n is a partition number A000041 then the rotated structure with n regions shows each row as a partition of k such that A000041(k) = n (see example).
For the definition of "regions of n" see A206437.

Examples

			For n = 11 the three views of the shell model of partitions with 11 regions look like this:
.
.     A182181(11) = 35           A210692(11) = 29
.
.   1                                       1
.   1                                       1
.   1                                       1
.   1                                       1
.   1       1                             1 1
.   1       1                             1 1
.   1       1   1                       1 1 1
.   2       1   1                       1 1 2
.   2       1   1   1                 1 1 1 2
.   3   2   2   2   1 1             1 1 2 2 3
.   6 3 4 2 5 3 4 2 3 2 1         1 2 3 4 5 6
. <------- Regions ------         ------------> N
.                            L
.                            a    1
.                            r    * 2
.                            g    * * 3
.                            e    * 2
.                            s    * * * 4
.                            t    * * 3
.                                 * * * * 5
.                            p    * 2
.                            a    * * * 4
.                            r    * * 3
.                            t    * * * * * 6
.                            s
.
.                                A182727(11) = 35
.
The areas of the shadows of the three views are A182181(11) = 35, A182727(11) = 35 and A210692(11) = 29, therefore the total area of the three shadows is 35+35+29 = 99, so a(11) = 99.
Since n = 11 is a partition number A000041 we can see that the rotated structure with 11 regions shows each row as a partition of 6 because A000041(6) = 11. See below:
.
.                      6
.                    3   3
.                  4       2
.                2   2       2
.              5               1
.            3   2               1
.          4       1               1
.        2   2       1               1
.      3       1       1               1
.    2   1       1       1               1
.  1   1   1       1       1               1
.

Crossrefs

Other versions: A207380, A210970, A210979, A210980, A210990.
Cf. A000041, A026905, A135010, A138121, A141285, A182703, A194446, A182181, A182727, A186114, A206437, A210692.

Formula

a(n) = A182181(n) + A182727(n) + A210692(n).
a(A000041(n)) = 2*A006128(n) + A026905(n).

A207379 Triangle read by rows: T(n,k) = number of parts that are in the k-th column of the last section of the set of partitions of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 1, 4, 4, 3, 2, 1, 1, 4, 4, 4, 3, 2, 1, 1, 7, 7, 6, 5, 3, 2, 1, 1, 8, 8, 8, 6, 5, 3, 2, 1, 1, 12, 12, 11, 10, 7, 5, 3, 2, 1, 1, 14, 14, 14, 12, 10, 7, 5, 3, 2, 1, 1, 21, 21, 20, 18, 14, 11, 7, 5, 3, 2, 1, 1
Offset: 1

Views

Author

Omar E. Pol, Mar 10 2012

Keywords

Comments

Note that for n >= 2 the tail of the last section of n starts at the second column and the second column contains only one part of size 1, thus both the first and the second columns contain the same number of parts. For more information see A135010 and A182703.

Examples

			Illustration of initial terms. First six rows of triangle as numbers of parts in the columns from the last sections of the first six natural numbers:
.                                       6
.                                       3 3
.                                       4 2
.                                       2 2 2
.                           5             1
.                           3 2             1
.                 4           1             1
.                 2 2           1             1
.         3         1           1             1
.   2       1         1           1             1
1     1       1         1           1             1
---------------------------------------------------
1,  1,1,  1,1,1,  2,2,1,1,  2,2,2,1,1,  4,4,3,2,1,1
...
Triangle begins:
1;
1,   1;
1,   1,  1;
2,   2,  1,  1;
2,   2,  2,  1,  1;
4,   4,  3,  2,  1,  1;
4,   4,  4,  3,  2,  1,  1;
7,   7,  6,  5,  3,  2,  1,  1;
8,   8,  8,  6,  5,  3,  2,  1,  1;
12, 12, 11, 10,  7,  5,  3,  2,  1,  1;
14, 14, 14, 12, 10,  7,  5,  3,  2,  1,  1;
21, 21, 20, 18, 14, 11,  7,  5,  3,  2,  1,  1;

Crossrefs

Column 1 is A187219. Row sums give A138137. Reversed rows converge to A000041.
Cf. A002865, A058399, A135010, A138135, A181187, A207031, A207380.

A210765 Triangle read by rows in which row n lists the number of partitions of n together with n-1 ones.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 5, 1, 1, 1, 7, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 15, 1, 1, 1, 1, 1, 1, 22, 1, 1, 1, 1, 1, 1, 1, 30, 1, 1, 1, 1, 1, 1, 1, 1, 42, 1, 1, 1, 1, 1, 1, 1, 1, 1, 56, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 77, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 101, 1, 1, 1
Offset: 1

Views

Author

Omar E. Pol, Mar 26 2012

Keywords

Comments

The sum of row n is S_n = n - 1 + A000041(n) = A133041(n) - 1.
Also consider a vertical rectangle on the infinite square grid with shorter side = n and longer side = p(n) = A000041(n). Each row of rectangle represents a partition of n. Each part of each partition of n is a horizontal rectangle with shorter side = 1 and longer side = k, where k is the size of the part. It appears that T(n,k) is also the number of k-th parts of all partitions of n in the k-th column of rectangle.

Examples

			Triangle begins:
1;
2,  1;
3,  1, 1;
5,  1, 1, 1;
7,  1, 1, 1, 1;
11, 1, 1, 1, 1, 1;
15, 1, 1, 1, 1, 1, 1;
22, 1, 1, 1, 1, 1, 1, 1;
30, 1, 1, 1, 1, 1, 1, 1, 1;
42, 1, 1, 1, 1, 1, 1, 1, 1, 1;

Crossrefs

Main diagonal of A209655 and of A209918.
Cf. A000041, A008284, A058399, A133041, A135010, A141285, A209656, A207380.
Showing 1-9 of 9 results.

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