A213476 - OEIS

A213476

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 7, n >= 2.

%I #9 Jun 29 2012 14:50:57

%S 2,4,6,10,14,20,26,18,2,2,4,10,22,38,60,110,138,188,106,108,54,36,4,2,

%T 4,10,22,50,104,194,300,444,542,840,650,1056,808,1144,354,292,16,2,4,

%U 10,22,50,104,234,460,778,894,1540,1812,3444,3512,8294,6104,13914,5778,5548,2216,710,24

%N 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 7, 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...18....19...20...21...22...23...24

%C .n

%C .2....2....4....6...10...14...20...26...18....2

%C .3....2....4...10...22...38...60..110..138..188..106..108...54...36....4

%C .4....2....4...10...22...50..104..194..300..444..542..840..650.1056..808.1144..354...292...16

%C .5....2....4...10...22...50..104..234..460..778..894.1540.1812.3444.3512.8294.6104.13914.5778.5548.2216..710...24

%C where k is the path length in nodes. There is insufficient evidence to attempt to define the irregularity of the array. However, the maximum values of k for 2 <= n <= 7 are 11, 16, 20, 24, 29, 33. Reading this array by rows gives the sequence. 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 >= k-1 is 2, 4, 10, 22, 50, 104 for which there appears to be no obvious formula.

%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>

%e 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 7 node rectangle.

%Y Cf. A213106, A213249, A213431, A213433, A213473, A213474, A213475.

%K nonn,tabf

%O 2,1

%A _Christopher Hunt Gribble_, Jun 12 2012