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%I #5 Jul 11 2012 13:51:53
%S 2,5,0,10,0,18,0,0,31,0,0,52,0,0,0,86,0,0,0,141,0,0,0,0,230,0,0,0,0,
%T 374,0,0,0,0,0,607,0,0,0,0,0,984,0,0,0,0,0,0,1594,0,0,0,0,0,0,2581,0,
%U 0,0,0,0,0,0,4178,0,0,0,0,0,0,0,6762,0,0,0,0,0,0
%N Irregular array T(n,k) of the numbers of non-extendable (complete) non-self-adjacent simple paths ending at each of a minimal subset of nodes within a square lattice bounded by rectangles with nodal dimensions n and 2, 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 1 to capture all geometrically distinct counts. The quarter-rectangle is read by rows. The irregular array of numbers is:
%C ....k.....1..2..3..4..5..6..7..8..9.10
%C ..n
%C ..2.......2
%C ..3.......5..0
%C ..4......10..0
%C ..5......18..0..0
%C ..6......31..0..0
%C ..7......52..0..0..0
%C ..8......86..0..0..0
%C ..9.....141..0..0..0..0
%C .10.....230..0..0..0..0
%C .11.....374..0..0..0..0..0
%C .12.....607..0..0..0..0..0
%C .13.....984..0..0..0..0..0..0
%C .14....1594..0..0..0..0..0..0
%C .15....2581..0..0..0..0..0..0..0
%C .16....4178..0..0..0..0..0..0..0
%C .17....6762..0..0..0..0..0..0..0..0
%C .18...10943..0..0..0..0..0..0..0..0
%C .19...17708..0..0..0..0..0..0..0..0..0
%C .20...28654..0..0..0..0..0..0..0..0..0
%C where k indicates the position of the end node in the quarter-rectangle. For each n, the maximum value of k is floor((n+1)/2). 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>
%F Let T(n,k) denote an element of the irregular array then it appears that T(n,k) = A000045(n+3) - 3, n >= 2, k = 1 and T(n,k) = 0, n >= 2, k >= 2.
%e When n = 2, the number of times (NT) each node in the rectangle is the end node (EN) of a complete non-self-adjacent simple path is
%e EN 0 1
%e 2 3
%e NT 2 2
%e 2 2
%e To limit duplication, only the top left-hand corner 2 is stored in the sequence, i.e. T(2,1) = 2.
%Y Cf. A213106, A213249, A213274, A213478.
%K nonn,tabf
%O 2,1
%A _Christopher Hunt Gribble_, Jul 04 2012
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