1,3

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.

Andrew Howroyd, Table of n, a(n) for n = 1..500

Index entries for linear recurrences with constant coefficients, signature (10,-40,80,-80,32).

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

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.

(PARI) a(n) = if(n==2, 1, n*(n-1)*(n^2+n+36)*2^(n-8)/3) \\ Andrew Howroyd, Nov 27 2025

Column k=2 of A390909.

Cf. A001792, A308914, A332426, A360716, A360717.

Sequence in context: A318125 A151540 A226466 * A322938 A026544 A026527

Adjacent sequences: A363961 A363962 A363963 * A363965 A363966 A363967

nonn,easy

Ivaylo Kortezov, Jun 30 2023

a(1)-a(2) prepended by Andrew Howroyd, Nov 27 2025

approved