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A213433 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 3, n >= 2.

%I A213433 #17 Jun 29 2012 11:21:49
%S A213433 2,4,2,2,4,6,0,4,2,4,10,18,8,8,14,2,4,10,22,34,22,36,22,18,2,4,10,22,
%T A213433 38,56,68,80,58,34,24,2,2,4,10,22,38,60,110,138,188,106,108,54,36,4,2,
%U A213433 4,10,22,38,60,114,188,280,360,248,254,174,84,52,6,2,4,10,22,38,60,114,192,338,494,694,534,642,402,282,130,72,8
%N A213433 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 3, n >= 2.
%C A213433 The irregular array of numbers is:
%C A213433 ...k..3...4...5...6...7...8...9..10..11..12..13..14..15..16..17..18..19..20
%C A213433 .n
%C A213433 .2....2...4...2
%C A213433 .3....2...4...6...0...4
%C A213433 .4....2...4..10..18...8...8..14
%C A213433 .5....2...4..10..22..34..22..36..22..18
%C A213433 .6....2...4..10..22..38..56..68..80..58..34..24...2
%C A213433 .7....2...4..10..22..38..60.110.138.188.106.108..54..36...4
%C A213433 .8....2...4..10..22..38..60.114.188.280.360.248.254.174..84..52...6
%C A213433 .9....2...4..10..22..38..60.114.192.338.494.694.534.642.402.282.130..72...8
%C A213433 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 2n+1 for 2 <= n <= 6 and 2n+2 for n >= 7. Reading this array by rows gives the sequence.
%C A213433 The asymptotic sequence for the number of distinct shapes under rotation of the complete non-self-adjacent simple paths of each nodal length k, where n >= k-1, is 2, 4, 10, 22, 38, 60, 114, 192, 342, 564, 956, 1584, 2686, 4524, 7684, 12968 for which there appears to be no obvious formula.
%H A213433 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 A213433 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>
%e A213433 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 3 node rectangle.
%Y A213433 Cf. A213106, A213249, A213431.
%K A213433 nonn,tabf
%O A213433 2,1
%A A213433 _Christopher Hunt Gribble_, Jun 11 2012
%E A213433 Added new comment.