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"Geodesic coherent" partition of a graph

Pilipczuk and Siebertz proved that every planar graph has such a partition with an even stronger property. Namely, each part $V_i$ is a geodesic path, and the graph obtained by contracting each part ...
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Partition of $(2^{n+1}+1)2^{2^{n-1}+n-1}-1$ into parts with binary weight equals $2^{n-1}+n$

Notice that for $i\in\{0,1,\dots,2^{n-1}+n\}$ we have $$a(i+1,2^{n-1}+n) = 2^{2^{n-1}+n+1} - 1 - 2^{2^{n-1}+n-i}.$$ Then the sum in question can be easily computed: \begin{split} & a(1,2^{n-1}+n)+\...

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