Assume that $q=2^s$ for some non-negative integer $s$. Is there any simple formula for:
$$\sum\limits_{i = 0}^s {\left( {\begin{array}{*{20}{c}} q \\ {{2^i}} \end{array}} \right)}?$$
2026-04-05 19:24:30.1775417070
Evaluation of $\sum\limits_{i = 0}^s {\left( {\begin{array}{*{20}{c}} q \\ {{2^i}} \end{array}} \right)}$
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I doubt it. I have computed the values for $1\le s\le 10$. Here are the first five: $$ 3, 11, 107, 14827, 611635179 $$ There is no sequence matching them in the OEIS.