On the upper bound for the density of any injective vector

In this work, the Stern's sequence $b_1 = 1,$ $b_2 = 1,$ $b_3 = 2,$ $b_4 = 3,$ $b_5 = 6,$ $b_6 = 11,$ $b_7 = 20,$ $b_8 = 40, \ldots$ is considered, and the upper and lower bounds for $b_i$ are determined. Supposing that the vector $(a_1, \ldots, a_r)$, where $r \geq 4,$ $a_1 = b_r$, $a_2 = b_r + b_{r - 1}$, $\ldots$, $a_r = \sum\limits_{i = 1}^r b_i$, is the injective one having the least maximum element among all other injective vectors of length $r$, the upper bound for density of any injective vector is stated.