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 A242522 Number of cyclic arrangements of S={1,2,...,n} such that the difference between any two neighbors is at least 2. 18
 0, 0, 0, 0, 1, 5, 33, 245, 2053, 19137, 196705, 2212037, 27029085, 356723177, 5058388153, 76712450925, 1239124984693, 21241164552785, 385159565775633, 7365975246680597, 148182892455224845, 3128251523599365177, 69149857480654157545, 1597343462243140957757 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS a(n)=NPC(n;S;P) is the count of all neighbor-property cycles for a specific set S of n elements and a specific pair-property P. For more details, see the link and A242519. Number of Hamiltonian cycles in the complement of P_n, where P_n is the n-path graph. - Andrew Howroyd, Mar 15 2016 LINKS Andrew Woods, Table of n, a(n) for n = 1..100 (terms up to a(24) from Hiroaki Yamanouchi, Aug 28 2014) F. C. Holroyd and W. J. G. Wingate, Cycles in the complement of a tree or other graph, Discrete Math., 55 (1985), 267-282. S. Sykora, On Neighbor-Property Cycles, Stan's Library, Volume V, 2014. Eric Weisstein's World of Mathematics, Hamiltonian Cycle Eric Weisstein's World of Mathematics, Path Complement Graph FORMULA a(n) = A002493(n)/(2*n), n>1. - Andrew Woods, Dec 08 2014 a(n) = Sum_{k=1..n} (-1)^(n-k)*k!*A102413(n,k) / (2*n), n>2. - Andrew Woods after Vladeta Jovovic, Dec 08 2014 a(n) = (n-1)!/2 + sum_{i=1..n-1} ((-1)^i * (n-i-1)! * sum_{j=0..i-1} (2^j * C(i-1,j) * C(n-i,j+1))), for n>=5. - Andrew Woods, Jan 08 2015 a(n) = n a(n-1) - (n-5) a(n-2) - (n-4) a(n-3) + (n-4) a(n-4), for n>6. - Jean-François Alcover, Oct 07 2017 EXAMPLE The 5 cycles of length n=6 having this property are {1,3,5,2,4,6}, {1,3,5,2,6,4}, {1,3,6,4,2,5}, {1,4,2,5,3,6}, {1,4,2,6,3,5}. MATHEMATICA a[n_ /; n < 5] = 0; a[5] = 1; a[6] = 5; a[n_] := a[n] = n a[n-1] - (n-5) a[n-2] - (n-4) a[n-3] + (n-4) a[n-4]; Array[a, 24] (* Jean-François Alcover, Oct 07 2017 *) Join[{0, 0}, RecurrenceTable[{a[n] == n a[n - 1] - (n - 5) a[n - 2] - (n - 4) a[n - 3] + (n - 4) a[n - 4], a[3] == a[4] == 0, a[5] == 1, a[6] == 5}, a, {n, 20}]] (* Eric W. Weisstein, Apr 12 2018 *) PROG (C++) See the link. CROSSREFS Cf. A242519, A242520, A242521, A242523, A242524, A242525, A242526, A242527, A242528, A242529, A242530, A242531, A242532, A242533, A242534. Cf. A006184. Sequence in context: A084131 A084771 A153398 * A034015 A268563 A056159 Adjacent sequences:  A242519 A242520 A242521 * A242523 A242524 A242525 KEYWORD nonn AUTHOR Stanislav Sykora, May 27 2014 STATUS approved

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Last modified November 19 08:53 EST 2018. Contains 317347 sequences. (Running on oeis4.)