A174469 Number of permutations p of {1,...,n} satisfying p(1)=1 and, if n>1, |p(i)-p((i mod n)+1)| is in {2,3} for i from 1 to n.

1, 0, 0, 0, 2, 0, 0, 0, 0, 2, 2, 2, 2, 2, 4, 6, 8, 10, 12, 16, 22, 30, 40, 52, 68, 90, 120, 160, 212, 280, 370, 490, 650, 862, 1142, 1512, 2002, 2652, 3514, 4656, 6168, 8170, 10822, 14336, 18992, 25160, 33330, 44152, 58488, 77480, 102640, 135970

Offset: 1

Author

Alois P. Heinz, Nov 28 2010

Keywords

• nonn
• easy

Comments

Also the number of directed Hamiltonian cycles in the graph on n vertices {1,...,n}, with i adjacent to j iff 2 <= |i-j| <= 3.

Examples

			For n = 10 the a(10) = 2 permutations are (1,3,6,9,7,10,8,5,2,4), (1,4,2,5,8,10,7,9,6,3).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Hamiltonian Cycle
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1).

Crossrefs

Cf. A017899.

Programs

Maple

a:= n-> `if`(n<2, n, (Matrix (5, (i,j)-> `if`(j-i=1 or i=5 and j in {1,5}, 1, 0))^n. <<2, -2, (0$3)>>)[1,1]): seq(a(n), n=1..60);

Mathematica

Join[{1}, LinearRecurrence[{1, 0, 0, 0, 1}, {0, 0, 0, 2, 0}, 60]] (* Jean-François Alcover, Oct 28 2021 *)

Formula

G.f.: (3*x^5-2*x^4+x-1)*x / (x^5+x-1).

a(n) = 2*A017899(n-5) for n>=5.