A213106 - OEIS

%I #31 Jun 14 2012 12:32:42

%S 4,10,32,20,82,276,36,198,898,4028,62,456,2770,16840,93664,104,1014,

%T 8098,65998,483974,3248120,172,2210,22886,250152,2430726,21169866,

%U 177690360,282,4758,63366,931076,12062348,136925026,1482885382,15972807764

%N 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.

%C The first 6 rows of the triangle are:

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

%C .n

%C .2.......4

%C .3......10....32

%C .4......20....82...276

%C .5......36...198...898...4028

%C .6......62...456..2770..16840...93664

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

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

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

%H C. H. Gribble, <a href="https://oeis.org/wiki/Complete_non-self-adjacent_paths:Results_for_Square_Lattice">Computed characteristics of complete non-self-adjacent paths in a square lattice bounded by various sizes of rectangle.</a>

%H C. H. Gribble, <a href="https://oeis.org/wiki/Complete non-self-adjacent paths:Program">Computes characteristics of complete non-self-adjacent paths in square and cubic lattices bounded by various sizes of rectangle and rectangular cuboid respectively.</a>

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

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

%F 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

%e 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.

%K nonn,tabl

%O 2,1

%A _Christopher Hunt Gribble_, Jun 05 2012