The n X n fiveleaper graph is connected for n >=8 and traceable from n = 8 up to at least n = 29, meaning a(n) = n^2 - 1 for a(n) with 8 <= n <= 29 (and probably all larger n).
Table of n, a(n) for n=1..29.
Eric Weisstein's World of Mathematics, Fiveleaper Graph
Eric Weisstein's World of Mathematics, Longest Path
Conjectures from Colin Barker, Jun 14 2019: (Start)
G.f.: x^5*(2 + 13*x - 4*x^2 - 23*x^3 + 13*x^4 + x^5) / (1 - x)^3.
a(n) = n^2 - 1 for n>7.
Cf. A307553 (number of longest paths).
Sequence in context: A109946 A141067 A296251 * A031911 A136685 A226489
Adjacent sequences: A307551 A307552 A307553 * A307555 A307556 A307557
Eric W. Weisstein, Apr 14 2019