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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.

%I A213249 #19 Jun 14 2012 19:18:07
%S A213249 2,8,16,18,64,134,34,170,706,1854,60,398,2346,13198,41478,102,880,
%T A213249 6832,55454,382116,1424988
%N 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.
%C A213249 The triangle of numbers is:
%C A213249 ....k....2....3.....4......5.......6........7
%C A213249 .n
%C A213249 .2.......2
%C A213249 .3.......8...16
%C A213249 .4......18...64...134
%C A213249 .5......34..170...706...1854
%C A213249 .6......60..398..2346..13198...41478
%C A213249 .7.....102..880..6832..55454..382116..1424988
%C A213249 The sequence is formed by reading the triangle by rows.
%H A213249 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 A213249 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 A213249 Let T(n,k) denote an element of the triangle then the following recurrence relations appear to hold:
%F A213249 T(n, 2) - T(n-1, 2) - 2*A000045(n+1) = 0, n >= 3,
%F A213249 T(n, 3) - 2*T(n-1, 3) - T(n-4, 3) - 4*(n+11) = 0, n >= 7.
%e A213249 T(2,2) = The number of rotationally distinct complete non-self-adjacent simple path shapes within a 2 X 2 node rectangle.
%Y A213249 Cf. A213106.
%K A213249 nonn,tabl
%O A213249 2,1
%A A213249 _Christopher Hunt Gribble_, Jun 07 2012