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 1-4 of 4 results.

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).

A210952 Triangle read by rows: T(n,k) = sum of all parts of the k-th column of the partitions of n but with the partitions aligned to the right margin.

Original entry on oeis.org

1, 1, 3, 1, 3, 5, 1, 3, 7, 9, 1, 3, 7, 12, 12, 1, 3, 7, 14, 21, 20, 1, 3, 7, 14, 24, 31, 25, 1, 3, 7, 14, 26, 40, 47, 38, 1, 3, 7, 14, 26, 43, 61, 66, 49, 1, 3, 7, 14, 26, 45, 70, 92, 93, 69, 1, 3, 7, 14, 26, 45, 73, 106, 130, 124, 87, 1, 3, 7, 14
Offset: 1

Views

Author

Omar E. Pol, Apr 22 2012

Keywords

Examples

			For n = 6 the illustration shows the partitions of 6 aligned to the right margin and below the sums of the columns:
.
.                      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,  3,  7, 14, 21, 20
.
So row 6 lists 1, 3, 7, 14, 21, 20.
Triangle begins:
1;
1, 3;
1, 3, 5;
1, 3, 7,  9;
1, 3, 7, 12, 12;
1, 3, 7, 14, 21, 20;
1, 3, 7, 14, 24, 31, 25;
1, 3, 7, 14, 26, 40, 47, 38;
1, 3, 7, 14, 26, 43, 61, 66, 49;
1, 3, 7, 14, 26, 45, 70, 92, 93, 69:

Crossrefs

Mirror of triangle A206283. Rows sums give A066186. Rows converge to A014153. Right border gives A046746, >= 1.
Cf. A135010, A138121, A181187, A210763, A210950, A210951, A210953, A210961, A210970.

Formula

T(n,k) = Sum_{j=1..n} A210953(j,k). - Omar E. Pol, May 26 2012

A210763 Tetrahedron T(j,n,k) in which the slice j is a finite triangle read by rows T(n,k) which lists the sums of the columns of the shell model of partitions with n shells.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 2, 2, 3, 1, 1, 2, 5, 1, 1, 2, 2, 2, 3, 2, 2, 3, 5, 1, 1, 1, 2, 7, 1, 1, 2, 2, 2, 3, 3, 3, 3, 5, 3, 3, 4, 4, 7, 1, 1, 1, 2, 4, 11, 1, 1, 2, 2, 2, 3, 3, 3, 3, 5, 4, 4, 5, 4, 7, 3, 3, 3, 5, 6, 11, 1, 1, 1, 1, 2, 4, 15
Offset: 1

Views

Author

Omar E. Pol, Apr 24 2012

Keywords

Examples

			--------------------------------------------------------
Illustration of first five                      A210952
slices of the tetrahedron                       Row sum
--------------------------------------------------------
. 1,                                               1
.    1,                                            1
.    1, 2,                                         3
.          1,                                      1
.          1, 2,                                   3
.          1, 1, 3,                                5
.                   1,                             1
.                   1, 2,                          3
.                   2, 2, 3,                       7
.                   1, 1, 2, 5,                    9
.                               1,                 1
.                               1, 2,              3
.                               2, 2, 3,           7
.                               2, 2, 3, 5,       12
.                               1, 1, 1, 2, 7,    12
--------------------------------------------------------
. 1, 2, 2, 3, 3, 3, 5, 5, 5, 5, 7, 7, 7, 7, 7,
Each column sum in the slice j is equal to A000041(j).
.
Also this sequence can be written as a triangle read by rows in which each row is a flattened triangle. The sequence begins:
1;
1,1,2;
1,1,2,1,1,3;
1,1,2,2,2,3,1,1,2,5;
1,1,2,2,2,3,2,2,3,5,1,1,1,2,7;
1,1,2,2,2,3,3,3,3,5,3,3,4,4,7,1,1,1,2,4,11;
1,1,2,2,2,3,3,3,3,5,4,4,5,4,7,3,3,3,5,6,11,1,1,1,1,2,4,15;
Row n has length A000217(n). Row sums give A066186. Right border gives A000041(n), n >= 1.

Crossrefs

Cf. A135010, A138121, A209655, A209918, A210952, A210960, A210961.

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.
Showing 1-4 of 4 results.

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