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URL: https://oeis.org/A363964

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A363964
Number of unordered pairs of disjoint self-avoiding paths with nodes that cover all vertices of a convex labeled n-gon; one-node paths are allowed.
2
0, 1, 3, 14, 55, 195, 644, 2016, 6048, 17520, 49280, 135168, 362752, 955136, 2472960, 6307840, 15876096, 39481344, 97124352, 236584960, 571146240, 1367539712, 3249799168, 7669284864, 17983078400, 41916825600, 97165246464, 224076496896, 514272002048, 1174992322560
OFFSET
1,3
COMMENTS
For each such path there is a sequence of distinct vertices of the n-gon, each (except the last one) connected by a segment with the next vertex in the sequence; the segments have no common internal points. The path itself is the union of the set of these segments and is thus direction-independent: reversing the order of the vertices leads to the same path. If the sequence of vertices has length 1 then there are no segments; we call such a path a singleton.
FORMULA
a(n) = n*(n-1)*(n^2+n+36)*2^(n-8)/3 for n != 2.
G.f.: x^2*(1 - x)*(1 - 3*x + 3*x^2)*(1 - 3*x + 6*x^2)/(1 - 2*x)^5. - Andrew Howroyd, Nov 14 2025
EXAMPLE
a(4)=14 since if one of the paths is a singleton (4 choices), then there are A001792(3)=3 choices for the other path, and otherwise for the two paths there are A308914(4)=2 choices, so a(4)=4*3+2=14.
PROG
(PARI) a(n) = if(n==2, 1, n*(n-1)*(n^2+n+36)*2^(n-8)/3) \\ Andrew Howroyd, Nov 27 2025
CROSSREFS
Column k=2 of A390909.
Sequence in context: A318125 A151540 A226466 * A322938 A026544 A026527
KEYWORD
nonn,easy
AUTHOR
Ivaylo Kortezov, Jun 30 2023
EXTENSIONS
a(1)-a(2) prepended by Andrew Howroyd, Nov 27 2025
STATUS
approved