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.

A225542 Number T(n,k,u) of partitions of an n X k rectangle into integer-sided square parts containing u nodes that are unconnected to any of their neighbors, considering only the number of parts; 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, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 2, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 0, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 0, 1, 0, 0, 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.

Examples

			The irregular triangle begins:
n,k\u 0   1   2   3   4   5   6   7   8   9  10  11  12 ...
1,1   1
2,1   1
2,2   1   1
3,1   1
3,2   1   1
3,3   1   1   0   0   1
4,1   1
4,2   1   1   1
4,3   1   1   1   0   1
4,4   1   1   1   1   2   0   0   0   0   1
5,1   1
5,2   1   1   1
5,3   1   1   1   0   1   1
5,4   1   1   1   1   2   1   1   0   0   1
5,5   1   1   1   1   2   1   1   1   0   1   0   0   0 ...
...
For n = 5 and k = 4 there are 2 partitions that contain 4 isolated nodes, so T(5,4,4) = 2.
Consider that each partition 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.  Then the 2 partitions are:
1 1 1 1 1    1 1 1 1 1
1 0 1 0 1    1 0 0 1 1
1 1 1 1 1    1 0 0 1 1
1 0 1 0 1    1 1 1 1 1
1 1 1 1 1    1 1 1 1 1
1 1 1 1 1    1 1 1 1 1

Links

Crossrefs

Cf. A034295, A224697, A227009, A225777, A225803, A225568.

Formula

T(n,n,u) = A227009(n,u).
Sum_{u=1..(n-1)^2} T(n,n,u) = A034295(n).

A225777 Number T(n,k,u) of distinct tilings of an n X k rectangle using integer-sided square tiles 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, 4, 0, 0, 1, 1, 1, 3, 1, 1, 6, 4, 0, 2, 1, 9, 16, 8, 5, 0, 0, 0, 0, 1, 1, 1, 4, 3, 1, 8, 12, 0, 3, 4, 1, 12, 37, 34, 15, 12, 4, 0, 0, 2, 1, 16, 78, 140, 88, 44, 68, 32, 0, 4, 0, 0, 0, 0, 0, 0, 1
Offset: 1

Views

Author

Christopher Hunt Gribble, Jul 26 2013

Keywords

Comments

The number of entries per row is given by A225568.

Examples

			The irregular triangle begins:
n,k\u 0   1   2   3   4   5   6   7   8   9  10  11  12 ...
1,1   1
2,1   1
2,2   1   1
3,1   1
3,2   1   2
3,3   1   4   0   0   1
4,1   1
4,2   1   3   1
4,3   1   6   4   0   2
4,4   1   9  16   8   5   0   0   0   0   1
5,1   1
5,2   1   4   3
5,3   1   8  12   0   3   4
5,4   1  12  37  34  15  12   4   0   0   2
5,5   1  16  78 140  88  44  68  32   0   4   0   0   0 ...
...
For n = 4, k = 3, there are 4 tilings that contain 2 isolated nodes, so T(4,3,2) = 4. 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.  Then the 4 tilings are:
   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 1 0 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 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. A045846, A219924, A226997, A225542, A225803, A225568.

Formula

T(n,k,0) = 1, T(n,k,1) = (n-1)(k-1), T(n,k,2) = (n^2(k-1) - n(2k^2+5k-13) + (k^2+13k-24))/2.
Sum_{u=1..(n-1)^2} T(n,n,u) = A045846(n).

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).
Showing 1-3 of 3 results.

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