Some bijective results about the area of Schröder paths (Q1885021)

The paper considers the sequence \((a_i)_{i\geq 1}=1,3,7,17,41,\dots\), which is M2665 in the book of \textit{N. J. A. Sloane} and \textit{S. Plouffe} [The encyclopedia of integer sequences (Academic Press, San Diego, CA) (1995; Zbl 0845.11001)]. Among several combinatorial interpretations, the odd-indexed terms \(1,7,41,\dots\) are known to count the total area under elevated Schröder paths of given length. This paper provides a similar interpretation for the terms \(3,17,99,\dots\). The key step is a bijection between certain self-avoiding paths and grand Motzkin paths.