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Irregular array T(n,k) of the numbers of non-extendable (complete) non-self-adjacent simple paths incorporating each of a minimal subset of nodes within a square lattice bounded by rectangles with nodal dimensions n and 5, n >= 2.
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%I #7 Jul 23 2012 12:47:42

%S 40,42,40,188,209,204,210,228,204,820,1007,1058,1008,907,776,3426,

%T 4601,5076,4601,4104,3608,5076,3608,2608,13344,18726,21050,18302,

%U 17364,15896,21307,15275,11148,50036,71736,81276,69029,67670,63148,80263,61229,46550,82942,60116,44196

%N Irregular array T(n,k) of the numbers of non-extendable (complete) non-self-adjacent simple paths incorporating each of a minimal subset of nodes within a square lattice bounded by rectangles with nodal dimensions n and 5, n >= 2.

%C The subset of nodes is contained in the top left-hand quarter of the rectangle and has nodal dimensions floor((n+1)/2) and 3 to capture all geometrically distinct counts.

%C The quarter-rectangle is read by rows.

%C The irregular array of numbers is:

%C ...k.....1.....2.....3.....4.....5.....6.....7.....8.....9....10....11....12

%C .n

%C .2......40....42....40

%C .3.....188...209...204...210...228...204

%C .4.....820..1007..1058..1008...907...776

%C .5....3426..4601..5076..4601..4104..3608..5076..3608..2608

%C .6...13344.18726.21050.18302.17364.15896.21307.15275.11148

%C .7...50036.71736.81276.69029.67670.63148.80263.61229.46550.82942.60116.44196

%C where k indicates the position of a node in the quarter-rectangle.

%C For each n, the maximum value of k is 3*floor((n+1)/2).

%C Reading this array by rows gives the sequence.

%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 When n = 2, the number of times (NT) each node in the rectangle (N) occurs in a complete non-self-adjacent simple path is

%e N 0 1 2 3 4

%e 5 6 7 8 9

%e NT 40 42 40 42 40

%e 40 42 40 42 40

%e To limit duplication, only the top left-hand corner 40 and the 42 and 40 to its right are stored in the sequence,

%e i.e. T(2,1) = 40, T(2,2) = 42 and T(2,3) = 40.

%Y Cf. A213106, A213249, A213375, A214023, A214359, A214397, A214399, A214504, A214510

%K nonn,tabf

%O 2,1

%A _Christopher Hunt Gribble_, Jul 21 2012

%E Comment corrected by _Christopher Hunt Gribble_, Jul 22 2012