A093424 Number of two-dimensional burst patterns of size n, i.e., translation inequivalent subsets of the grid Z^2 which can be covered by a connected subset of n elements (in the sense of von Neumann neighborhoods).

1, 3, 13, 62, 306, 1546, 7958

Offset

1,2

Comments

Original definition: Number of different two-dimensional burst patterns in the grid graph: The grid graph has Z^2 as vertices and each vertex (x,y) is connected to (x-1,y),(x+1,y),(x,y-1),(x,y+1). A cluster of size t is a set of t points such that each pair of points of the set is on a connected path contained entirely within the set. A burst pattern is a labeling of Z^2 with 0's and 1's. The term a(n) denotes the number of different (up to a translation) burst patterns whose 1's are covered by a cluster of size n.

As shown in the examples 3, 5, 6 and 7 below, the set need not be connected itself. But this is always the case when it has n points; in such a case, it coincides with the covering set. - M. F. Hasler, Aug 28 2014

Links

Table of n, a(n) for n=1..7.

M. Blaum, J. Bruck, A. Vardy, Interleaving schemes for multidimensional cluster errors, IEEE Trans. on Inform. Theory, 44(2):730-743, March 1998.

Tuvi Etzion and Alexander Vardy, Two-dimensional interleaving schemes with repetitions: constructions and bounds, IEEE Trans. on Inform. Theory, 48(2):428-457, 2002.

Moshe Schwartz and Tuvi Etzion, Two-dimensional burst-correcting codes, Proceedings, International Symposium on Information Theory, 2004.

Example

a(3) = 13 because we have the following burst patterns (the *'s indicate the 1's):
1) *
2) **
3) *.*
4) *
...*
5) *
....
...*
6) *
....*
7) .*
...*
8) ***
9) **
....*
10) *
....**
11) .*
....**
12) **
....*
13) *
....*
....*
The absence of a V-shaped pattern, which would have a (taxicab) "diameter" of 3 but cannot be covered by a 3-element connected set, illustrates that the latter condition cannot be replaced by the former. - M. F. Hasler, Aug 28 2014

Crossrefs

Cf. A093426, A093427.

Sequence in context: A355987 A141786 A122122 * A186242 A367060 A350478

Adjacent sequences: A093421 A093422 A093423 * A093425 A093426 A093427

Keyword

nonn,more

Author

Tuvi Etzion and Moshe Schwartz (etzion(AT)cs.technion.ac.il), May 11 2004

Extensions

Definition simplified and example corrected by M. F. Hasler, Aug 20 2014

Status

approved