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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A346531 a(n) is the number of edges of the polycube called "tower" described in A221529 where n is the longest side of its base.

Original entry on oeis.org

12, 12, 27, 36, 51, 72, 84, 105, 117, 144, 165
Offset: 1

Views

Author

Omar E. Pol, Jul 22 2021

Keywords

Comments

The tower is a geometric object associated to all partitions of n.
The height of the tower equals A000041(n-1).

Examples

			For n = 1 the tower is a cube, and a cube has 12 edges, so a(1) = 12.

Crossrefs

Cf. A000203 (area of the terraces), A000041 (height of the terraces), A066186 (volume), A345023 (surface area), A346530 (number of faces), A346532 (number of vertices).
Cf. A325301 (analog for the pyramid described in A245092).
Cf. A221529, A236104, A237270, A237271, A237593, A336811, A338156, A340035, A340584.

Formula

a(n) = A346530(n) + A346532(n) - 2 (Euler's formula).

A346532 a(n) is the number of vertices of the polycube called "tower" described in A221529 where n is the longest side of its base.

Original entry on oeis.org

8, 8, 18, 24, 33, 47, 55, 69, 77, 95, 108
Offset: 1

Views

Author

Omar E. Pol, Jul 22 2021

Keywords

Comments

The height of the tower equals A000041(n-1).

Examples

			For n = 1 the tower is a cube, and a cube has 8 vertices, so a(1) = 8.

Crossrefs

Cf. A000203 (area of the terraces), A000041 (height of the terraces), A066186 (volume), A345023 (surface area), A346530 (number of faces), A346531 (number of edges).
Cf. A325302 (analog for the pyramid described in A245092).
Cf. A221529, A236104, A237270, A237271, A237593, A336811, A338156, A340035, A340584.

Formula

a(n) = A346531(n) - A346530(n) + 2 (Euler's formula).

A346562 Irregular triangle read by rows in which row n lists the first n - 2 terms of A000005 together with the sum of A000005(n-1) and A000005(n), with a(1) = 1.

Original entry on oeis.org

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

Views

Author

Omar E. Pol, Jul 23 2021

Keywords

Comments

T(n,k) is the total number of divisors related to the terraces that are in the k-th level that contains terraces starting from the base of the symmetric tower described in A221529.

Examples

			Triangle begins:
1;
3;
1, 4;
1, 2, 5;
1, 2, 2, 5;
1, 2, 2, 3, 6;
1, 2, 2, 3, 2, 6;
1, 2, 2, 3, 2, 4, 6;
1, 2, 2, 3, 2, 4, 2, 7;
1, 2, 2, 3, 2, 4, 2, 4, 7;
1, 2, 2, 3, 2, 4, 2, 4, 3, 6;
1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 8;
1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 8;
1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 6;
...

Crossrefs

The length of row n is A028310(n-1).
Row sums give A006218, n >= 1.
Leading diagonal gives A092405.
Other diagonals give A000005.
Column 1 gives the absolute values of A260196.
Companion of A346533.
Cf. A221529, A237270, A237593, A336811, A338156, A340035.

A350333 Irregular triangle read by rows in which row n lists all elements of the arrangement of the correspondence divisor/part related to the partitions of n in the following order: row n lists the n-th row of A026792 followed by the n-th row of A338156.

Original entry on oeis.org

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

Views

Author

Omar E. Pol, Dec 25 2021

Keywords

Examples

			Triangle begins:
[1], [1];
[2, 1, 1], [1, 2, 1];
[3, 2, 1, 1, 1, 1], [1, 3, 1, 2, 1, 1];
[4, 2, 2, 3, 1, 2, 1, 1, 1, 1, 1, 1], [1, 2, 4, 1, 3, 1, 2, 1, 2, 1, 1, 1];
...
Illustration of the first six rows of triangle in an infinite table:
.
|---|---------|-----|-------|---------|-----------|-------------|---------------|
| n |         |  1  |   2   |    3    |     4     |      5      |       6       |
|---|---------|-----|-------|---------|-----------|-------------|---------------|
|   |         |     |       |         |           |             |  6            |
| P |         |     |       |         |           |             |  3 3          |
| A |         |     |       |         |           |             |  4 2          |
| R |         |     |       |         |           |             |  2 2 2        |
| T |         |     |       |         |           |  5          |  5 1          |
| I |         |     |       |         |           |  3 2        |  3 2 1        |
| T |         |     |       |         |  4        |  4 1        |  4 1 1        |
| I |         |     |       |         |  2 2      |  2 2 1      |  2 2 1 1      |
| O |         |     |       |  3      |  3 1      |  3 1 1      |  3 1 1 1      |
| N |         |     |  2    |  2 1    |  2 1 1    |  2 1 1 1    |  2 1 1 1 1    |
| S |         |  1  |  1 1  |  1 1 1  |  1 1 1 1  |  1 1 1 1 1  |  1 1 1 1 1 1  |
----|---------|-----|-------|---------|-----------|-------------|---------------|
|   | A027750 |  1  |  1 2  |  1   3  |  1 2   4  |  1       5  |  1 2 3     6  |
|   | A027750 |     |  1    |  1 2    |  1   3    |  1 2   4    |  1       5    |
|   | A027750 |     |       |  1      |  1 2      |  1   3      |  1 2   4      |
|   | A027750 |     |       |  1      |  1 2      |  1   3      |  1 2   4      |
|   | A027750 |     |       |         |  1        |  1 2        |  1   3        |
| D | A027750 |     |       |         |  1        |  1 2        |  1   3        |
| I | A027750 |     |       |         |  1        |  1 2        |  1   3        |
| V | A027750 |     |       |         |           |  1          |  1 2          |
| I | A027750 |     |       |         |           |  1          |  1 2          |
| S | A027750 |     |       |         |           |  1          |  1 2          |
| O | A027750 |     |       |         |           |  1          |  1 2          |
| R | A027750 |     |       |         |           |  1          |  1 2          |
| S | A027750 |     |       |         |           |             |  1            |
|   | A027750 |     |       |         |           |             |  1            |
|   | A027750 |     |       |         |           |             |  1            |
|   | A027750 |     |       |         |           |             |  1            |
|   | A027750 |     |       |         |           |             |  1            |
|   | A027750 |     |       |         |           |             |  1            |
|   | A027750 |     |       |         |           |             |  1            |
|---|---------|-----|-------|---------|-----------|-------------|---------------|
.
For n = 6 in the upper zone of the above table we can see the partitions of 6 in reverse-colexicographic order in accordance with the 6th row of A026792.
In the lower zone of the table we can see the terms from the 6th row of A338156, these are the divisors of the numbers from the 6th row of A176206.
Note that in the lower zone of the table every row gives A027750.
The total number of rows in the table is equal to A000070(6+1) = 30.
The remarkable fact is that the elements in the lower zone of the arrangement are the same as the elements in the upper zone but in other order.
For an explanation of the connection of the elements of the upper zone with the elements of the lower zone, that is the correspondence divisor/part, see A338156.
For n = 10 we can see a representation of the upper zone (the partitions) and of the lower zone (the divisors) with the two polycubes described in A221529 respectively: a prism of partitions and a tower whose terraces are the symmetric representation of sigma(m), for m = 1..10. Each polycube has A066186(10) = 420 cubic cells, hence the total number of cubic cells is equal to A220909(10) = 840, equaling the sum of the 10th row of this triangle.

Crossrefs

Row sums give A220909.
Row lengths give A211978.
Cf. A350357 (analog for the last section of the set of partitions of n).
Cf. A000041, A000070, A006128, A026792, A027750, A066186, A135010, A176206, A221529, A237593, A336811, A336812, A338156, A340035.

A341148 Triangle read by rows: T(n,k) is number of cubes in the k-th vertical slice of the polycube called "tower" described in A221529 where n is the longest side of its base, 1 <= k <= n.

Original entry on oeis.org

1, 2, 2, 4, 3, 2, 7, 6, 4, 3, 12, 10, 7, 3, 3, 19, 17, 12, 9, 5, 4, 30, 26, 20, 13, 8, 4, 4, 45, 41, 31, 23, 16, 10, 5, 5, 67, 60, 48, 34, 25, 15, 11, 5, 5, 97, 89, 71, 55, 39, 28, 17, 12, 6, 6, 139, 127, 104, 78, 60, 40, 28, 17, 11, 6, 6, 195, 181, 149, 118, 89, 65, 45, 32, 21, 15, 7, 7
Offset: 1

Views

Author

Omar E. Pol, Feb 06 2021

Keywords

Comments

The row sums of triangle give A066186 because the correspondence divisor/part. For more information see A338156.
For further information about the tower see A221529.

Examples

			Triangle begins:
    1;
    2,   2;
    4,   3,   2;
    7,   6,   4,   3;
   12,  10,   7,   3,  3;
   19,  17,  12,   9,  5,  4;
   30,  26,  20,  13,  8,  4,  4;
   45,  41,  31,  23, 16, 10,  5,  5;
   67,  60,  48,  34, 25, 15, 11,  5,  5;
   97,  89,  71,  55, 39, 28, 17, 12,  6,  6;
  139, 127, 104,  78, 60, 40, 28, 17, 11,  6,  6;
  195, 181, 149, 118, 89, 65, 45, 32, 21, 15,  7,  7;
...
Illustration of initial terms:
              Top view
  n   k       of the tower       Heights        T(n,k)
               _
  1   1       |_|                1                 1
.              _ _
  2   1       |   |              1 1               2
  2   2       |_ _|              1 1               2
.              _ _ _
  3   1       |_|   |            2 1 1             4
  3   2       |    _|            1 1 1             3
  3   3       |_ _|              1 1               2
.              _ _ _ _
  4   1       |_| |   |          3 2 1 1           7
  4   2       |_ _|   |          2 2 1 1           6
  4   3       |      _|          1 1 1 1           4
  4   4       |_ _ _|            1 1 1             3
.              _ _ _ _ _
  5   1       |_| | |   |        5 3 2 1 1        12
  5   2       |_ _|_|   |        3 3 2 1 1        10
  5   3       |_ _|  _ _|        2 2 1 1 1         7
  5   4       |     |            1 1 1             3
  5   5       |_ _ _|            1 1 1             3
.              _ _ _ _ _ _
  6   1       |_| | | |   |      7 5 3 2 1 1      19
  6   2       |_ _|_| |   |      5 5 3 2 1 1      17
  6   3       |_ _|  _|   |      3 3 2 2 1 1      12
  6   4       |_ _ _|    _|      2 2 2 1 1 1       9
  6   5       |        _|        1 1 1 1 1         5
  6   6       |_ _ _ _|          1 1 1 1           4
.
The levels of the terraces of the tower are the partition numbers A000041 starting from the base.
Note that the top view of the tower is essentially the same as the top view of the stepped pyramid described in A245092 except that in the tower both the symmetric representation of sigma(n) and the symmetric representation of sigma(n-1) are unified in the level 1 of the structure because the first two partitions numbers A000041 are [1, 1].

Crossrefs

Column 1 gives A000070.
Leading diagonal gives A080513.
Row sums give A066186.
Cf. A000041, A024916, A236104, A237270, A237271, A237593, A221529, A245092, A262626, A336811, A336812, A338156, A340035.
Previous Showing 21-25 of 25 results.

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