A214504 Irregular array T(n,k) of the numbers of non-extendable (complete) non-self-adjacent simple paths incorporating each of a minimal subset of nodes within a square lattice bounded by rectangles with nodal dimensions n and 3, n >= 2.

12, 14, 32, 36, 36, 48, 80, 88, 86, 100, 188, 210, 209, 228, 204, 204, 418, 470, 472, 524, 479, 452, 906, 1016, 1028, 1152, 1050, 1020, 1088, 980, 1943, 2170, 2219, 2472, 2250, 2222, 2333, 2200, 4137, 4610, 4754, 5260, 4811, 4738, 4929, 4784, 4920, 4924

Offset

2,1

Comments

The subset of nodes is contained in the top left-hand quarter of the rectangle and has nodal dimensions floor((n+1)/2) and 2 to capture all geometrically distinct counts.

The quarter-rectangle is read by rows.

The irregular array of numbers is:

....k....1....2....3....4....5....6....7....8....9...10

..n

..2.....12...14

..3.....32...36...36...48

..4.....80...88...86..100

..5....188..210..209..228..204..204

..6....418..470..472..524..479..452

..7....906.1016.1028.1152.1050.1020.1088..980

..8...1943.2170.2219.2472.2250.2222.2333.2200

..9...4137.4610.4754.5260.4811.4738.4929.4784.4920.4924

where k indicates the position of a node in the quarter-rectangle.

For each n, the maximum value of k is 2*floor((n+1)/2).

Reading this array by rows gives the sequence.

Links

Table of n, a(n) for n=2..49.

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

When n = 2, the number of times (NT) each node in the rectangle (N) occurs in a complete non-self-adjacent simple path is
N   0  1  2
    3  4  5
NT 12 14 12
   12 14 12
To limit duplication, only the top left-hand corner 12 and the 14 to its right are stored in the sequence,
  i.e. T(2,1) = 12 and T(2,2) = 14.

Crossrefs

Cf. A213106, A213249, A213089, A213954, A214121, A214397, A214399

Sequence in context: A022326 A307167 A238228 * A140810 A330197 A127401

Adjacent sequences: A214501 A214502 A214503 * A214505 A214506 A214507

Keyword

nonn,tabf

Author

Christopher Hunt Gribble, Jul 19 2012

Extensions

Comment corrected by Christopher Hunt Gribble, Jul 22 2012

Status

approved