This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A213274 #37 Dec 29 2012 15:19:36 %S A213274 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 A213274 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 A213274 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 A213274 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 A213274 The irregular array of numbers is: %C A213274 ....k..3...4...5...6...7...8...9..10..11..12..13..14..15..16..17 %C A213274 ..n %C A213274 ..2....4 %C A213274 ..3....4...4...2 %C A213274 ..4....4...4...6...6 %C A213274 ..5....4...4...6..10..10...2 %C A213274 ..6....4...4...6..10..14..16...8 %C A213274 ..7....4...4...6..10..14..20..26..18...2 %C A213274 ..8....4...4...6..10..14..20..30..40..34..10 %C A213274 ..9....4...4...6..10..14..20..30..44..60..60..28...2 %C A213274 .10....4...4...6..10..14..20..30..44..64..90.100..62..12 %C A213274 .11....4...4...6..10..14..20..30..44..64..94.134.160.122..40...2 %C A213274 where k is the path length in nodes. %C A213274 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 A213274 Reading this array by rows gives the sequence. %C A213274 One half of the numbers of paths constitute the sequence to remove the effect of the bilateral symmetry of a rectangle. %H A213274 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 A213274 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 A213274 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 A213274 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 A213274 Cf. A213106, A213249. %K A213274 nonn,tabf %O A213274 2,1 %A A213274 _Christopher Hunt Gribble_, Jun 08 2012