A213106 Triangle T(n,k) of numbers/2 of non-extendable (complete) non-self-adjacent simple paths within a square lattice bounded by rectangles with nodal dimensions n and k, n >= k >= 2.

4, 10, 32, 20, 82, 276, 36, 198, 898, 4028, 62, 456, 2770, 16840, 93664, 104, 1014, 8098, 65998, 483974, 3248120, 172, 2210, 22886, 250152, 2430726, 21169866, 177690360, 282, 4758, 63366, 931076, 12062348, 136925026, 1482885382, 15972807764

Offset: 2

Author

Christopher Hunt Gribble, Jun 05 2012

Keywords

• nonn
• tabl

Comments

The first 6 rows of the triangle are:

....k....2.....3.....4......5.......6........7

.n

.2.......4

.3......10....32

.4......20....82...276

.5......36...198...898...4028

.6......62...456..2770..16840...93664

.7.....104..1014..8098..65998..483974..3248120

Reading this triangle by rows gives the first 21 terms of the sequence.

One half of the numbers of paths constitute the sequence to remove the effect of the bilateral symmetry of a rectangle.

Examples

			T(2,2) = One half of the number of complete non-self-adjacent simple paths within a square lattice bounded by a 2 X 2 node rectangle.
		

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.

Formula

Let T(n,k) denote an element of the triangle then the following recurrence relations appear to hold:

T(n, 2) - T(n-1, 2) - 2*A000045(n+1) = 0, n >= 3

T(n, 3) - 3*T(n-1, 3) + 2*T(n-2, 3) - T(n-4, 3) + T(n-5, 3) - 8*(n-4) = 0, n >= 9