$b[ $i ] = $a[ $i ] + $a[ --$i ] while $i;

Except that crashes with a "Modification of non-creatable array value attempted ..." fatality (after a couple of warnings) for an empty input array. I've only visually examined them, but all the preceeding for-loop and map (and even grep!) solutions seem as if they would handle empty arrays gracefully. But
    $b[ $i ] = $a[ $i ] + $a[ --$i ] while $i > 0;
avoids the problem.