A213473 Irregular array T(n,k) of the numbers of distinct shapes under rotation of the non-extendable (complete) non-self-adjacent simple paths of each length within a square lattice bounded by rectangles with nodal dimensions n and 4, n >= 2.

2, 4, 6, 6, 2, 4, 10, 18, 8, 8, 14, 2, 4, 10, 18, 20, 12, 18, 8, 42, 2, 4, 10, 22, 46, 66, 60, 56, 106, 72, 236, 26, 2, 4, 10, 22, 50, 100, 152, 158, 230, 246, 410, 260, 546, 124, 32, 2, 4, 10, 22, 50, 104, 194, 300, 444, 542, 840, 650, 1056, 808, 1144, 354, 292, 16

Offset

2,1

Comments

The irregular array of numbers is:

...k..3....4....5....6....7....8....9...10...11...12...13...14...15...16...17...18...19...20

.n

.2....2....4....6....6

.3....2....4...10...18....8....8...14

.4....2....4...10...18...20...12...18....8...42

.5....2....4...10...22...46...66...60...56..106...72..236...26

.6....2....4...10...22...50..100..152..158..230..246..410..260..546..124...32

.7....2....4...10...22...50..104..194..300..444..542..840..650.1056..808.1144..354..292...16

where k is the path length in nodes. There is insufficient evidence to attempt to define the irregularity of the array. However, the maximum values of k for 2 <= n <= 12 are 6, 9, 11, 14, 17, 20, 22, 25, 28, 30, 33. Reading this array by rows gives the sequence. The asymptotic sequence for the number of distinct shapes under rotation of the complete non-self-adjacent simple paths of each nodal length k, where n >= k-1, is 2, 4, 10, 22, 50, 104, 198, 354, 710, 1288, 2600 for which there appears to be no obvious formula.

Links

Table of n, a(n) for n=2..66.

C. H. Gribble, Computed characteristics of complete non-self-adjacent paths in a square lattice bounded by various sizes of rectangle.

C. H. Gribble, Computes characteristics of complete non-self-adjacent paths in square and cubic lattices bounded by various sizes of rectangle and rectangular cuboid respectively.

Example

T(2,3) = The number of distinct shapes under rotation of the complete non-self-adjacent simple paths of length 3 nodes within a square lattice bounded by a 2 X 4 node rectangle.

Crossrefs

Cf. A213106, A213249, A213431, A213433.

Sequence in context: A234574 A010587 A372035 * A134920 A011031 A248844

Adjacent sequences: A213470 A213471 A213472 * A213474 A213475 A213476

Keyword

nonn,tabf

Author

Christopher Hunt Gribble, Jun 12 2012

Status

approved