[seqfan] A005207+1 and A296516 as numbers of terms in some polynomials

[Luc Rousseau]

Dears SeqFans,

I'm studying two sequences u(n) and v(n) (n >= 1) defined by
u(n) = the number of terms in the expanded form of bivariate polynomial P_n,
v(n) = the number of terms in the expanded form of bivariate polynomial Q_n,

where P_n and Q_n are defined by
P_1 = P = x + y,
Q_1 = Q = x * y,
P_{n+1} = P(P_n, Q_n),
Q_{n+1} = Q(P_n, Q_n).

Example: Q_2 = (P_1) * (Q_1) = (x + y) * (xy) = x^2*y + x*y^2. Two terms ==> v(2) = 2.

I am making no progress towards proving (or refuting) the conjectured formulas:
u(n) = (F(2*n-1) + F(n+1))/2 + 1,
v(n) = (F(2*n+1) + F(n+2))/2 - T(n-1) - 1,
where F(n) denotes the n-th Fibonacci number and T(n) the n-th triangular number.
Does anyone see a demonstration?

Proved, the result would deserve:
- a comment in long-existing A005207 : a(n) = u(n) - 1 = the number of "+" in P_n expanded,
- a complete reformulation of A296516(n) =def v(n) (draft status),
- crossref A005207 <-> A296516.

Kind Regards,

Luc Rousseau.