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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.

%I A264813 #20 Feb 16 2025 08:33:27
%S A264813 1,0,1,1,0,3,6,0,53,199,0,2908,13699,0,369985,2135430,0,87265700,
%T A264813 611286653,0
%N 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.
%C A264813 a(n) = 0 for n == 1 (mod 3).
%H A264813 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LangfordsProblem.html">Langford's Problem</a>
%H A264813 Wikipedia, <a href="https://en.wikipedia.org/wiki/Dancing_Links">Dancing Links</a>
%H A264813 Wikipedia, <a href="https://en.wikipedia.org/wiki/Langford_pairing">Langford pairing</a>
%e A264813 a(0) = 1: the empty permutation.
%e A264813 a(2) = 1: 221121.
%e A264813 a(3) = 1: 223321131.
%e A264813 a(5) = 3: 223325534411514, 225523344531141, 552244253341131.
%e A264813 a(6) = 6: 221121665544336543, 225523366534411614, 225526633544361141, 446611415563322532, 552266253344631141, 665544336543221121.
%Y A264813 Cf. A014552, A104185, A108235, A176127, A203435, A261516, A261517, A321956.
%K A264813 nonn,more
%O A264813 0,6
%A A264813 _Alois P. Heinz_, Nov 25 2015