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 1-3 of 3 results.

A224850 Number T(n,k) of tilings of an n X k rectangle using integer-sided square tiles reduced for symmetry, where the orbits under the symmetry group of the rectangle, D2, have 1 element; triangle T(n,k), k >= 1, 0 <= n < k, read by columns.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 2, 3, 1, 1, 5, 2, 12, 6, 1, 1, 3, 3, 5, 7, 17, 1, 1, 8, 3, 25, 11, 106, 44
Offset: 1

Views

Author

Christopher Hunt Gribble, Jul 22 2013

Keywords

Comments

It appears that sequence T(2,k) consists of 2 interspersed Fibonacci sequences.
The diagonal T(n,n) is A006081. - M. F. Hasler, Jul 25 2013

Examples

			The triangle is:
n\k  1   2   3   4   5   6   7   8 ...
.
0    1   1   1   1   1   1   1   1 ...
1        1   1   1   1   1   1   1 ...
2            1   3   2   5   3   8 ...
3                1   2   2   3   3 ...
4                    3  12   5  25 ...
5                        6   7  11 ...
6                           17 106 ...
7                               44 ...
...
T(3,5) = 2 because there are 2 different tilings of the 3 X 5 rectangle by integer-sided squares, where any sequence of group D2 operations will only transform each tiling into itself.  Group D2 operations are:
.   the identity operation
.   rotation by 180 degrees
.   reflection about a horizontal axis through the center
.   reflection about a vertical axis through the center
The tilings are:
._________.    ._________.
|_|_|_|_|_|    |_|     |_|
|_|_|_|_|_|    |_|     |_|
|_|_|_|_|_|    |_|_____|_|

Links

Crossrefs

Cf. A219924, A224697, A227690.

Formula

T(n,k) + A224861(n,k) + A224867(n,k) = A227690(n,k).
1*T(n,k) + 2*A224861(n,k) + 4*A224867(n,k) = A219924(n,k).

A224867 Number T(n,k) of tilings of an n X k rectangle using integer-sided square tiles reduced for symmetry, where the orbits under the symmetry group of the rectangle, D2, have 4 elements; triangle T(n,k), k >= 1, 0 <= n < k, read by columns.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 5, 21, 0, 0, 0, 10, 65, 440, 0, 0, 0, 27, 222, 1901, 14508, 0, 0, 0, 58, 676, 7716, 81119, 856559
Offset: 1

Views

Author

Christopher Hunt Gribble, Jul 22 2013

Keywords

Examples

			The triangle is:
n\k    1      2      3      4      5      6      7      8 ...
.
0      0      0      0      0      0      0      0      0 ...
1             0      0      0      0      0      0      0 ...
2                    0      0      0      0      0      0 ...
3                           1      5     10     27     58 ...
4                                 21     65    222    676 ...
5                                       440   1901   7716 ...
6                                            14508  81119 ...
7                                                  856559 ...
...
T(3,5) = 5 because there are 5 different sets of 4 tilings of the 3 X 5 rectangle by integer-sided squares, where any sequence of group D2 operations will transform each tiling in a set into another in the same set.  Group  D2 operations are:
.   the identity operation
.   rotation by 180 degrees
.   reflection about a horizontal axis through the center
.   reflection about a vertical axis through the center
An example of a tiling in each set is:
._________.  ._________.  ._________.  ._________.  ._________.
|   |_|_|_|  |_|   |_|_|  |   |   |_|  |   |_|_|_|  |   |     |
|_ _|_|_|_|  |_|_ _|_|_|  |_ _|_ _|_|  |___|   |_|  |___|     |
|_|_|_|_|_|  |_|_|_|_|_|  |_|_|_|_|_|  |_|_|___|_|  |_|_|_____|

Links

Crossrefs

Cf. A219924, A224697, A227690.

Formula

A224850(n,k) + A224861(n,k) + T(n,k) = A227690(n,k).
1*A224850(n,k) + 2*A224861(n,k) + 4*T(n,k) = A219924(n,k).

A225803 Number T(n,k,u) of tilings of an n X k rectangle using integer-sided square tiles, reduced for symmetry, containing u nodes that are unconnected to any of their neighbors; irregular triangle T(n,k,u), 1 <= k < n, u >= 0, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 0, 1, 1, 1, 2, 2, 1, 2, 4, 0, 2, 1, 1, 4, 13, 10, 6, 3, 1, 0, 0, 1, 1, 1, 3, 4, 1, 1, 3, 8, 3, 2, 3, 0, 0, 1, 1, 6, 23, 33, 24, 15, 6, 0, 2, 2, 2, 1, 1, 6, 40, 101, 129, 79, 74, 53, 13, 9, 11, 4, 0, 0, 0, 0, 1
Offset: 1

Views

Author

Christopher Hunt Gribble, Jul 28 2013

Keywords

Comments

The number of entries per row is given by A225568(n>0 and n != A000217(1:)).

Examples

			The irregular triangle T(n,k,u) begins:
n,k\u  0   1   2   3   4   5   6   7   8   9  10  11  12 ...
2,1    1
3,1    1
3,2    1   1
4,1    1
4,2    1   2   1
4,3    1   2   2   0   1
5,1    1
5,2    1   2   2
5,3    1   2   4   0   2   1
5,4    1   4  13  10   6   3   1   0   0   1
6,1    1
6,2    1   3   4   1
6,3    1   3   8   3   2   3   0   0   1
6,4    1   6  23  33  24  15   6   0   2   2   1
6,5    1   6  40 101  79  74  53  13   9  11   4   0   0 ...
...
T(5,3,2) = 4 because there are 4 different sets of tilings of the 5 X 3 rectangle by integer-sided squares in which each tiling contains 2 isolated nodes.  Any sequence of group D2 operations will transform each tiling in a set into another in the same set.  Group D2 operations are:
.   the identity operation
.   rotation by 180 degrees
.   reflection about a horizontal axis through the center
.   reflection about a vertical axis through the center
A 2 X 2 square contains 1 isolated node.  Consider that each tiling is composed of ones and zeros where a one represents a node with one or more links to its neighbors and a zero represents a node with no links to its neighbors.  An example of a tiling in each set is:
   1 1 1 1    1 1 1 1    1 1 1 1    1 1 1 1
   1 0 1 1    1 0 1 1    1 0 1 1    1 0 1 1
   1 1 1 1    1 1 1 1    1 1 1 1    1 1 1 1
   1 0 1 1    1 1 0 1    1 1 1 1    1 1 1 1
   1 1 1 1    1 1 1 1    1 0 1 1    1 1 0 1
   1 1 1 1    1 1 1 1    1 1 1 1    1 1 1 1

Links

Crossrefs

Cf. A224239, A227004, A227690, A224850, A224861, A224867, A225777, A225542.

Formula

T1(n,k,0) = 1, T1(n,k,1) = floor(n/2)*floor(k/2).
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