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%I #37 Dec 29 2012 15:19:36
%S 4,4,4,2,4,4,6,6,4,4,6,10,10,2,4,4,6,10,14,16,8,4,4,6,10,14,20,26,18,
%T 2,4,4,6,10,14,20,30,40,34,10,4,4,6,10,14,20,30,44,60,60,28,2,4,4,6,
%U 10,14,20,30,44,64,90,100,62,12,4,4,6,10,14,20,30,44,64,94,134,160,122,40,2
%N Irregular array T(n,k) of numbers/2 of non-extendable (complete) non-self-adjacent simple paths of each length within a square lattice bounded by rectangles with nodal dimensions n and 2, n >= 2.
%C The irregular array of numbers is:
%C ....k..3...4...5...6...7...8...9..10..11..12..13..14..15..16..17
%C ..n
%C ..2....4
%C ..3....4...4...2
%C ..4....4...4...6...6
%C ..5....4...4...6..10..10...2
%C ..6....4...4...6..10..14..16...8
%C ..7....4...4...6..10..14..20..26..18...2
%C ..8....4...4...6..10..14..20..30..40..34..10
%C ..9....4...4...6..10..14..20..30..44..60..60..28...2
%C .10....4...4...6..10..14..20..30..44..64..90.100..62..12
%C .11....4...4...6..10..14..20..30..44..64..94.134.160.122..40...2
%C where k is the path length in nodes.
%C In an attempt to define the irregularity of the array, it appears that the maximum value of k is (3n + n mod 2)/2 for n >= 2.
%C Reading this array by rows gives 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 The asymptotic sequence for the number of paths of each nodal length k for n >> 0 appears to be 2*A097333(1:), that is, 2*(Sum(j=0..k-2, C(k-2-j, floor(j/2)))), for k >= 3.
%e T(2,3) = One half of the number of complete non-self-adjacent simple paths of length 3 nodes within a square lattice bounded by a 2 X 2 node rectangle.
%Y Cf. A213106, A213249.
%K nonn,tabf
%O 2,1
%A _Christopher Hunt Gribble_, Jun 08 2012
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