OFFSET
0,5
COMMENTS
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 {2,3,4}.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..500
W. Edwin Clark, permutations p in S_n such that m <= |p(i)-p(i+1)| <= M for i from 1 to n-1, SeqFan Discussion, Mar 27 2010.
EXAMPLE
For n = 5 the a(5) = 14 permutations are (1,3,5,2,4), (1,4,2,5,3), (2,4,1,3,5), (2,4,1,5,3), (2,5,3,1,4), (3,1,4,2,5), (3,1,5,2,4), (3,5,1,4,2), (3,5,2,4,1), (4,1,3,5,2), (4,2,5,1,3), (4,2,5,3,1), (5,2,4,1,3), (5,3,1,4,2).
MAPLE
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(2, 4, n): seq(a(n), n=1..12); # Alois P. Heinz, Mar 27 2010
MATHEMATICA
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[2, 4, n]; Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 12}] (* Jean-François Alcover, Jun 01 2015, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
W. Edwin Clark, Mar 27 2010
EXTENSIONS
Edited by Alois P. Heinz, Nov 27 2010
a(22) from Alois P. Heinz, Oct 12 2013
a(23) from Alois P. Heinz, Jan 14 2016
a(24)-a(35) from Andrew Howroyd, Apr 05 2016
STATUS
approved