Average of near-diagonal numbers

HardCombinatorics Brainteasers~9m

Consider all $10$ -digit numbers $\overline{d_9 d_8 \cdots d_1 d_0}$ (so $d_9 \ne 0$ ) whose digits satisfy $|d_i - i| \le 1$ for every position $i \in \{0, 1, \dots, 9\}$ . What is the average of all such numbers?

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The constraint is per-position, so valid numbers form a product set - average each digit independently, weighted by $10^i$ .
Interior digits average to $i$ ; the boundaries clip to $\bar d_0 = 0.5$ and $\bar d_9 = 8.5$ .

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Average of near-diagonal numbers

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