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A174707
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The number of permutations p of {1,...,n} such that |p(i)-p(i+1)| is in {3,4,5} for all i from 1 to n-1.
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13
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1, 0, 0, 0, 0, 2, 28, 144, 292, 272, 160, 272, 844, 3888, 15830, 49080, 113468, 208224, 352112, 662810, 1497286, 3853054, 10238142, 25892602, 60223752, 130042700, 271136524, 572265830, 1258121046, 2878870324, 6714840216, 15583281118, 35434903508, 78777769972, 172664047056
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OFFSET
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1,6
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COMMENTS
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For n>1, a(n)/2 is the number of Hamiltonian paths on the graph with vertex set {1,...,n} where i is adjacent to j iff |i-j| is in {3,4,5}.
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LINKS
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EXAMPLE
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For n = 6 the a(6) = 2 permutations are (3,6,2,5,1,4), (4,1,5,2,6,3).
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MAPLE
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f:= proc(m, M, n) option remember; local i, l, p, cnt; l:= array([i$i=1..n]); cnt:=0; p:= proc(t) local d, j, h; if t=n then d:= `if`(t=1, m, abs(l[t]-l[t-1])); if m<=d and d<=M then cnt:= cnt+1 fi else for j from t to n do l[t], l[j]:= l[j], l[t]; d:= `if`(t=1, m, abs(l[t]-l[t-1])); if m<=d and d<=M then p(t+1) fi od; h:= l[t]; for j from t to n-1 do l[j]:= l[j+1] od; l[n]:= h fi end; p(1); cnt end: a:= n-> f(3, 5, n); seq(a(n), n=1..14); # Alois P. Heinz, Mar 27 2010
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MATHEMATICA
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f[m_, M_, n_] := f[m, M, n] = Module[{i, l, p, cnt}, Do[l[i] = i, {i, 1, n}]; cnt = 0; p[t_] := Module[{d, j, h}, If[t == n, d = If[t == 1, m, Abs[l[t] - l[t-1]]]; If [m <= d && d <= M, cnt = cnt+1], For[j = t, j <= n, j++, {l[t], l[j]} = {l[j], l[t]}; d = If[t == 1, m, Abs[l[t] - l[t-1]]]; If [m <= d && d <= M, p[t+1]]]; h = l[t]; For[j = t, j <= n-1, j++, l[j] = l[j+1]]; l[n] = h]]; p[1]; cnt]; a[n_] := f[3, 5, n]; Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 14}] (* Jean-François Alcover, Jun 01 2015, after Alois P. Heinz *)
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CROSSREFS
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Cf. A003274, A174700, A174701, A174702, A174703, A174704, A174705, A174706, A174708, A185030, A216837.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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