A214119 Irregular array T(n,k) of the numbers of non-extendable (complete) non-self-adjacent simple paths ending at each of a minimal subset of nodes within a square lattice bounded by rectangles with nodal dimensions n and 2, n >= 2.

2, 5, 0, 10, 0, 18, 0, 0, 31, 0, 0, 52, 0, 0, 0, 86, 0, 0, 0, 141, 0, 0, 0, 0, 230, 0, 0, 0, 0, 374, 0, 0, 0, 0, 0, 607, 0, 0, 0, 0, 0, 984, 0, 0, 0, 0, 0, 0, 1594, 0, 0, 0, 0, 0, 0, 2581, 0, 0, 0, 0, 0, 0, 0, 4178, 0, 0, 0, 0, 0, 0, 0, 6762, 0, 0, 0, 0, 0, 0

Offset: 2

Author

Christopher Hunt Gribble, Jul 04 2012

Keywords

• nonn
• tabf

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 1 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.......2

..3.......5..0

..4......10..0

..5......18..0..0

..6......31..0..0

..7......52..0..0..0

..8......86..0..0..0

..9.....141..0..0..0..0

.10.....230..0..0..0..0

.11.....374..0..0..0..0..0

.12.....607..0..0..0..0..0

.13.....984..0..0..0..0..0..0

.14....1594..0..0..0..0..0..0

.15....2581..0..0..0..0..0..0..0

.16....4178..0..0..0..0..0..0..0

.17....6762..0..0..0..0..0..0..0..0

.18...10943..0..0..0..0..0..0..0..0

.19...17708..0..0..0..0..0..0..0..0..0

.20...28654..0..0..0..0..0..0..0..0..0

where k indicates the position of the end node in the quarter-rectangle. For each n, the maximum value of k is floor((n+1)/2). Reading this array by rows gives the sequence.

Examples

			When n = 2, the number of times (NT) each node in the rectangle is the end node (EN) of a complete non-self-adjacent simple path is
EN 0 1
   2 3
NT 2 2
   2 2
To limit duplication, only the top left-hand corner 2 is stored in the sequence, i.e. T(2,1) = 2.
		

Links

• 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.

Crossrefs

Cf. A213106, A213249, A213274, A213478.

Formula

Let T(n,k) denote an element of the irregular array then it appears that T(n,k) = A000045(n+3) - 3, n >= 2, k = 1 and T(n,k) = 0, n >= 2, k >= 2.