A213249 Triangle T(n,k) of numbers of distinct shapes under rotation 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.

2, 8, 16, 18, 64, 134, 34, 170, 706, 1854, 60, 398, 2346, 13198, 41478, 102, 880, 6832, 55454, 382116, 1424988

Offset

2,1

Comments

The triangle of numbers is:

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

.n

.2.......2

.3.......8...16

.4......18...64...134

.5......34..170...706...1854

.6......60..398..2346..13198...41478

.7.....102..880..6832..55454..382116..1424988

The sequence is formed by reading the triangle by rows.

Links

Table of n, a(n) for n=2..22.

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) - 2*T(n-1, 3) - T(n-4, 3) - 4*(n+11) = 0, n >= 7.

Example

T(2,2) = The number of rotationally distinct complete non-self-adjacent simple path shapes within a 2 X 2 node rectangle.

Crossrefs

Cf. A213106.

Sequence in context: A174882 A080095 A193219 * A155853 A256552 A031061

Adjacent sequences: A213246 A213247 A213248 * A213250 A213251 A213252

Keyword

nonn,tabl

Author

Christopher Hunt Gribble, Jun 07 2012

Status

approved