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.
The 6th rows of A225777 and A225542, that is, when n = 3 and k = 3, both contain 5 entries, so a(6) = 5.
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
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
T(4,3,2,2) = 4 because the 4 X 3 X 2 rectangular cuboid can be dissected in 4 distinct ways in which there are 2 nodes unconnected to any of their neighbors. The dissections and isolated nodes can be illustrated by expanding into 2 dimensions:
._______. ._______. ._______.
| | | | . | . | | | |
|___|___| |___|___| |___|___|
|_|_|_|_| |_|_|_|_| |_|_|_|_|
._______. ._______. ._______.
| |_|_| | . |_|_| | |_|_|
|___| | |___| . | |___| |
|_|_|___| |_|_|___| |_|_|___|
._______. ._______. ._______.
|_|_| | |_|_| . | |_|_| |
| |___| | . |___| | |___|
|___|_|_| |___|_|_| |___|_|_|
._______. ._______. ._______.
|_|_|_|_| |_|_|_|_| |_|_|_|_|
| | | | . | . | | | |
|___|___| |___|___| |___|___|
.
The irregular triangle begins:
u 0 1 2 3 4 5 6 7 8 9 10 11 12 ...
n k r
1,1,1 1
2,1,1 1
2,2,1 1
2,2,2 1 1
3,1,1 1
3,2,1 1
3,2,2 1 2
3,3,1 1
3,3,2 1 4
3,3,3 1 8 0 0 0 0 0 0 1
4,1,1 1
4,2,1 1
4,2,2 1 3 1
4,3,1 1
4,3,2 1 6 4
4,3,3 1 12 16 0 0 0 0 0 2
4,4,1 1
4,4,2 1 9 16 8 1
4,4,3 1 18 64 64 16 0 0 0 4
4,4,4 1 27 193 544 707 454 142 20 9 0 0 0 0 ...
Comments