Let me think of it. But just in passing. I checked with the
Encyclopedia of Integer Sequences (https://www.research.att.com/~njas/sequences/) for 2, 7, 19, 38 ... The sequence has not been recorded. So that if you are not mistaken as regard 19 and 38, you have a chance to register a new sequences under your name.

I'll be thinking of your problem.

Alexander Bogomolny

I am not sure about that. I think that they are about the same. You may get a more straightforward problem if, in addition, you also exclude direct moves from Dst to Aux.

With those three restrictions you may be able to obtain an upper bound for your probems.

The problem with three restrictions may be looked at this way. Place your pegs on a circle so that only clockwise moves are allowed. The pegs Src, Aux, Dst will be traversed clockwise. Let's forget for a moment about the designations source, auxiliary and destination, and think of just having N disks on one of the pegs. Let T_n be the number of moves needed to move the pegs 1 arc clockwise, and S_n be the number of moves needed to move n pegs 1 arc counterclockwise. Then

T_n = S_n-1 + 1 + S_n-1, and
S_n = T_n-1 + 2 + T_n-1.

In matrix notations, let v_n = (T_n S_n)^T and a = (1 2)^T. Then

v_n = A·v_n-1 + a, where A is the matrix with rows (0 2) and (2 0).

We may take v₀ = 0, so that v₁ = a. Then, successively,

v₂ = Aa + a,
v₃ = A²a + Aa + a,
v₄ = A³a + A²a + Aa + a, etc.

The solution is almost immediate if you separate the sequences of odd and even n's.