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A213954 Irregular array T(n,k) of the numbers of non-extendable (complete) non-self-adjacent simple paths starting at each of a minimal subset of nodes within a square lattice bounded by rectangles with nodal dimensions n and 3, n >= 2.

%I A213954 #20 Oct 16 2018 11:24:01
%S A213954 3,4,8,6,6,8,17,14,12,10,36,32,25,18,20,12,77,68,51,36,38,20,164,142,
%T A213954 106,72,72,38,64,28,347,298,225,146,142,74,109,46,732,628,476,302,294,
%U A213954 148,197,82,168,64,1543,1324,1003,632,614,304,385,156,277,100,3252,2790,2112,1328,1284,634,777,312,504,174,414,136
%N A213954 Irregular array T(n,k) of the numbers of non-extendable (complete) non-self-adjacent simple paths starting at each of a minimal subset of nodes within a square lattice bounded by rectangles with nodal dimensions n and 3, n >= 2.
%C A213954 The subset of nodes approximately defines the top left-hand quarter of the rectangle and has nodal dimensions floor((n+1)/2) and 2 to capture all geometrically distinct counts.
%C A213954 The quarter-rectangle is read by rows.
%C A213954 The irregular array of numbers is:
%C A213954 ....k.....1....2....3....4....5....6....7....8....9...10...11...12
%C A213954 ..n
%C A213954 ..2.......3....4
%C A213954 ..3.......8....6....6....8
%C A213954 ..4......17...14...12...10
%C A213954 ..5......36...32...25...18...20...12
%C A213954 ..6......77...68...51...36...38...20
%C A213954 ..7.....164..142..106...72...72...38...64...28
%C A213954 ..8.....347..298..225..146..142...74..109...46
%C A213954 ..9.....732..628..476..302..294..148..197...82..168...64
%C A213954 .10....1543.1324.1003..632..614..304..385..156..277..100
%C A213954 .11....3252.2790.2112.1328.1284..634..777..312..504..174..414..136
%C A213954 where k indicates the position of the start node in the quarter-rectangle.
%C A213954 For each n, the maximum value of k is 2*floor((n+1)/2).
%C A213954 Reading this array by rows gives the sequence.
%H A213954 C. H. Gribble, <a href="/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 A213954 C. H. Gribble, <a href="/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 A213954 It appears that:
%F A213954 T(n,1) - 2*T(n-1,1) - T(n-4,1) - 2 = 0, n >= 6
%F A213954 T(n,2) - 2*T(n-1,2) - T(n-4,1) = 0, n >= 6
%F A213954 T(n,3) - 2*T(n-1,3) - T(n-4,1) = 0, n >= 10
%F A213954 T(n,4) - 2*T(n-1,4) - T(n-4,1) + 8 = 0, n >= 7
%e A213954 When n = 2, the number of times (NT) each node in the rectangle is the start node (SN) of a complete non-self-adjacent simple path is
%e A213954 SN 0 1 2
%e A213954    3 4 5
%e A213954 NT 3 4 3
%e A213954    3 4 3
%e A213954 To limit duplication, only the top left-hand corner 3 and the 4 to its right are stored in the sequence, i.e. T(2,1) = 3 and T(2,2) = 4.
%Y A213954 Cf. A213106, A213249, A213089, A213478.
%K A213954 nonn,tabf
%O A213954 2,1
%A A213954 _Christopher Hunt Gribble_, Jun 30 2012