Get ready Part 15

Let $A(n)$ denote the number of sums of positive integers $a_1+a_2+\dots+a_r$ which add up to $n$ with $a_1>a_2+a_3;a_2>a_3 +a_4;\dots;a_{r-2}>a_{r-1}+a_r;a_{r-1}>a_r$ .Let $B(n)$ denote the number of $b_1+b_2+\dots+b_s$ which add up to $n$ , with

• $b_1 \ge b_2 \ge \dots \ge b_s$ ,

• Each $b_i$ is in the sequence $1,2,4,\dots,g_j,\dots$ defined by $g_1=1,g_2=2$ , and $g_j=g_{j-1}+g_{j-2}+1$ , and

• If $b_1=g_k$ then every element in $\{1,2,4,...,g_k\}$ appears at least once as a $b_i$ .

Is $A(n)=B(n)$ for each $n \ge 1$ ?