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 A249796 Triangle T(n,k), n>=3, 3<=k<=n, read by rows. Number of ways to make n selections without replacement from a circular array of n unlabeled cells (ignoring rotations and reflection), such that the first selection of a cell adjacent to previously selected cells occurs on the k-th selection. 2
 1, 1, 2, 2, 6, 4, 6, 18, 28, 8, 24, 72, 128, 120, 16, 120, 360, 672, 840, 496, 32, 720, 2160, 4128, 5760, 5312, 2016, 64, 5040, 15120, 29280, 43200, 47616, 32928, 8128, 128, 40320, 120960, 236160, 360000, 435264, 387072, 201728, 32640, 256, 362880, 1088640, 2136960, 3326400, 4249920, 4314240, 3121152, 1226880, 130816, 512 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS With m=n+3, T(m,3) = n!, T(m,m) = 2^n (easy proofs), and T(m,m-1) = A006516(n) = 2^(n-1) * (2^n - 1). Remaining supplied elements generated by exhaustive examination of permutations. LINKS EXAMPLE T(3,3) = 1 since, given any permutation of <1,2,3>, the third element will be the first to be adjacent to previous elements (modulo 3), and these 6 permutations are indistinguishable given rotations and reflection. Sample table (left-justified): .....1 .....1........2 .....2........6........4 .....6.......18.......28........8 ....24.......72......128......120.......16 ...120......360......672......840......496.......32 ...720.....2160.....4128.....5760.....5312.....2016.......64 ..5040....15120....29280....43200....47616....32928.....8128......128 .40320...120960...236160...360000...435264...387072...201728....32640......256 362880..1088640..2136960..3326400..4249920..4314240..3121152..1226880...130816......512 PROG (Sage) # Counting by exhaustive examination after a C program by Bartoletti. def A249796_row(n):     def F(p, n):         for k in range(2, n):             a = mod(p[k] + 1, n)             b = mod(p[k] - 1, n)             fa, fb = false, false             for i in range(k):                 if a == p[i] : fa = true                 if b == p[i] : fb = true             if fa and fb:                counts[k] += 1                return     counts = [0]*n     for p in Permutations(n):         F(p, n)     for k in range(2, n):         counts[k] = counts[k] / (2*n)     return counts for n in range(9): A249796_row(n) # Peter Luschny, Nov 11 2014 CROSSREFS Sequence in context: A222404 A081111 A092686 * A182411 A067804 A074911 Adjacent sequences:  A249793 A249794 A249795 * A249797 A249798 A249799 KEYWORD nonn,tabl AUTHOR Tony Bartoletti, Nov 05 2014 STATUS approved

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Last modified January 27 03:09 EST 2022. Contains 350601 sequences. (Running on oeis4.)