A264813 Number of permutations of 3 indistinguishable copies of 1,...,n such that the first and second copies of j are adjacent and there are exactly j numbers between the second and the third copy of j.

1, 0, 1, 1, 0, 3, 6, 0, 53, 199, 0, 2908, 13699, 0, 369985, 2135430, 0, 87265700, 611286653, 0, 34928617573, 284205647124, 0

Offset

0,6

Comments

a(n) = 0 for n == 1 (mod 3).

Links

Table of n, a(n) for n=0..22.

Eric Weisstein's World of Mathematics, Langford's Problem

Wikipedia, Dancing Links

Wikipedia, Langford pairing

Example

a(0) = 1: the empty permutation.
a(2) = 1: 221121.
a(3) = 1: 223321131.
a(5) = 3: 223325534411514, 225523344531141, 552244253341131.
a(6) = 6: 221121665544336543, 225523366534411614, 225526633544361141, 446611415563322532, 552266253344631141, 665544336543221121.

Crossrefs

Cf. A014552, A104185, A108235, A176127, A203435, A261516, A261517, A321956.

Sequence in context: A299032 A094674 A125287 * A009014 A009036 A021281

Adjacent sequences: A264810 A264811 A264812 * A264814 A264815 A264816

Keyword

nonn,more

Author

Alois P. Heinz, Nov 25 2015

Extensions

a(20)-a(22) from Martin Fuller, Jan 25 2026

Status

approved