Using the definition that a graph is "almost Hamiltonian" if its Hamiltonian length (length of a Hamiltonian walk) is one greater than the vertex count.
Table of n, a(n) for n=1..8.
Eric Weisstein's World of Mathematics, Almost Hamiltonian Graph
Eric Weisstein's World of Mathematics, Hamiltonian Walk
Eric Weisstein's World of Mathematics, Hamiltonian Number
Sequence in context: A355155 A058822 A054369 * A198028 A244300 A203209
Adjacent sequences: A185357 A185358 A185359 * A185361 A185362 A185363
Eric W. Weisstein, Aug 29 2013