A213954 Irregular array T(n,k) of the numbers of non-extendable (complete) non-self-adjacent simple paths starting at each of a minimal subset of nodes within a square lattice bounded by rectangles with nodal dimensions n and 3, n >= 2.

3, 4, 8, 6, 6, 8, 17, 14, 12, 10, 36, 32, 25, 18, 20, 12, 77, 68, 51, 36, 38, 20, 164, 142, 106, 72, 72, 38, 64, 28, 347, 298, 225, 146, 142, 74, 109, 46, 732, 628, 476, 302, 294, 148, 197, 82, 168, 64, 1543, 1324, 1003, 632, 614, 304, 385, 156, 277, 100, 3252, 2790, 2112, 1328, 1284, 634, 777, 312, 504, 174, 414, 136

Offset

2,1

Comments

The subset of nodes approximately defines 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...11...12

..n

..2.......3....4

..3.......8....6....6....8

..4......17...14...12...10

..5......36...32...25...18...20...12

..6......77...68...51...36...38...20

..7.....164..142..106...72...72...38...64...28

..8.....347..298..225..146..142...74..109...46

..9.....732..628..476..302..294..148..197...82..168...64

.10....1543.1324.1003..632..614..304..385..156..277..100

.11....3252.2790.2112.1328.1284..634..777..312..504..174..414..136

where k indicates the position of the start 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..71.

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.

Formula

It appears that:

T(n,1) - 2*T(n-1,1) - T(n-4,1) - 2 = 0, n >= 6

T(n,2) - 2*T(n-1,2) - T(n-4,1) = 0, n >= 6

T(n,3) - 2*T(n-1,3) - T(n-4,1) = 0, n >= 10

T(n,4) - 2*T(n-1,4) - T(n-4,1) + 8 = 0, n >= 7

Example

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

Crossrefs

Cf. A213106, A213249, A213089, A213478.

Sequence in context: A253080 A050417 A327613 * A357067 A074212 A125715

Adjacent sequences: A213951 A213952 A213953 * A213955 A213956 A213957

Keyword

nonn,tabf

Author

Christopher Hunt Gribble, Jun 30 2012

Status

approved