cp's OEIS Frontend

A213431 Irregular array T(n,k) of the numbers of distinct shapes under rotation of the 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.

%I A213431 #12 Jun 20 2012 13:43:53
%S A213431 2,2,4,2,2,4,6,6,2,4,6,10,10,2,2,4,6,10,14,16,8,2,4,6,10,14,20,26,18,
%T A213431 2,2,4,6,10,14,20,30,40,34,10,2,4,6,10,14,20,30,44,60,60,28,2,2,4,6,
%U A213431 10,14,20,30,44,64,90,100,62,12
%N A213431 Irregular array T(n,k) of the numbers of distinct shapes under rotation of the 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 A213431 The irregular array of numbers is:
%C A213431 ....k..3...4...5...6...7...8...9..10..11..12..13..14..15
%C A213431 ..n
%C A213431 ..2....2
%C A213431 ..3....2...4...2
%C A213431 ..4....2...4...6...6
%C A213431 ..5....2...4...6..10..10...2
%C A213431 ..6....2...4...6..10..14..16...8
%C A213431 ..7....2...4...6..10..14..20..26..18...2
%C A213431 ..8....2...4...6..10..14..20..30..40..34..10
%C A213431 ..9....2...4...6..10..14..20..30..44..60..60..28...2
%C A213431 .10....2...4...6..10..14..20..30..44..64..90.100..62..12
%C A213431 where k is the path length in nodes. In an attempt to define the irregularity of the array, it appears that the maximum value of k is n + floor((n+1)/2) for n >= 2. Reading this array by rows gives the sequence.
%H A213431 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 A213431 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 A213431 The asymptotic sequence for the number of distinct shapes under rotation of the complete non-self-adjacent simple paths of each nodal length k for n >> 0 appears to be 2*A097333(2:), that is, 2*(Sum(j=0..k-2, C(k-2-j, floor(j/2)))), for k >= 4.
%e A213431 T(2,3) = The number of distinct shapes under rotation of the complete non-self-adjacent simple paths of length 3 nodes within a square lattice bounded by a 2 X 2 node rectangle.
%Y A213431 Cf. A213106, A213249.
%K A213431 nonn,tabf
%O A213431 2,1
%A A213431 _Christopher Hunt Gribble_, Jun 11 2012