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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 2, n >= 2.
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%I #17 Jul 03 2012 15:56:17

%S 2,3,4,5,5,8,7,6,13,10,8,21,15,11,10,34,23,16,13,55,36,24,18,16,89,57,

%T 37,26,21,144,91,58,39,29,26,233,146,92,60,42,34,377,235,147,94,63,47,

%U 42,610,379,236,149,97,68,55,987,612,380,238,152,102,76,68,1597,989,613,382,241,157,110,89

%N 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 2, n >= 2.

%C The subset of nodes approximately defines the top left-hand quarter of the rectangle and has nodal dimensions floor((n+1)/2) and 1 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

%C ..n

%C ..2.......2

%C ..3.......3...4

%C ..4.......5...5

%C ..5.......8...7...6

%C ..6......13..10...8

%C ..7......21..15..11..10

%C ..8......34..23..16..13

%C ..9......55..36..24..18..16

%C .10......89..57..37..26..21

%C .11.....144..91..58..39..29..26

%C .12.....233.146..92..60..42..34

%C .13.....377.235.147..94..63..47..42

%C .14.....610.379.236.149..97..68..55

%C .15.....987.612.380.238.152.102..76..68

%C .16....1597.989.613.382.241.157.110..89

%C where k indicates the position of the start 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

%F T(n,k) = A000045(n-k+2), k = 0

%F T(n,k) = A000045(n-k+2) + A000045(k+1), k > 0.

%e 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 SN 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.

%K nonn,tabf

%O 2,1

%A _Christopher Hunt Gribble_, Jun 12 2012

%E Improved Comments