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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. 8
 3, 4, 8, 6, 6, 8, 17, 14, 12, 10, 36, 32, 25, 18, 20, 12, 77, 68, 51, 36, 38, 20, 164, 142, 106, 72, 72, 38, 64, 28, 347, 298, 225, 146, 142, 74, 109, 46, 732, 628, 476, 302, 294, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,1 COMMENTS 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. The quarter-rectangle is read by rows. The irregular array of numbers is: ....k.....1....2....3....4....5....6....7....8....9...10...11...12 ..n ..2.......3....4 ..3.......8....6....6....8 ..4......17...14...12...10 ..5......36...32...25...18...20...12 ..6......77...68...51...36...38...20 ..7.....164..142..106...72...72...38...64...28 ..8.....347..298..225..146..142...74..109...46 ..9.....732..628..476..302..294..148..197...82..168...64 .10....1543.1324.1003..632..614..304..385..156..277..100 .11....3252.2790.2112.1328.1284..634..777..312..504..174..414..136 where k indicates the position of the start node in the quarter-rectangle. For each n, the maximum value of k is 2*floor((n+1)/2). Reading this array by rows gives the sequence. LINKS Table of n, a(n) for n=2..71. C. H. Gribble, Computed characteristics of complete non-self-adjacent paths in a square lattice bounded by various sizes of rectangle. C. H. Gribble, Computes characteristics of complete non-self-adjacent paths in square and cubic lattices bounded by various sizes of rectangle and rectangular cuboid respectively. FORMULA It appears that: T(n,1) - 2*T(n-1,1) - T(n-4,1) - 2 = 0, n >= 6 T(n,2) - 2*T(n-1,2) - T(n-4,1) = 0, n >= 6 T(n,3) - 2*T(n-1,3) - T(n-4,1) = 0, n >= 10 T(n,4) - 2*T(n-1,4) - T(n-4,1) + 8 = 0, n >= 7 EXAMPLE 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 SN 0 1 2 3 4 5 NT 3 4 3 3 4 3 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. CROSSREFS Cf. A213106, A213249, A213089, A213478. Sequence in context: A253080 A050417 A327613 * A357067 A074212 A125715 Adjacent sequences: A213951 A213952 A213953 * A213955 A213956 A213957 KEYWORD nonn,tabf AUTHOR Christopher Hunt Gribble, Jun 30 2012 STATUS approved

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Last modified April 17 04:22 EDT 2024. Contains 371756 sequences. (Running on oeis4.)