Bender, Edward A.; Kochman, Fred; West, Douglas B. Adding up to powers. (English) Zbl 0742.11010 Am. Math. Mon. 97, No. 2, 139-143 (1990). The authors give an elegant combinatorial proof of the remarkable Moessner’s process which generates the well-known procedure of obtaining the squares by adding the odd numbers: \(1+3+\ldots+(2n-1)=n^2\). Start with the sequence \(1,2,3,\ldots\). For given \(k\), at \(j\)-th stage \((j<k)\) delete every \((k+1-j)\)-th member of the current sequence and construct the sequence of partial sums of the surviving sequence. Then the \((k-1)\)-th stage leaves the sequence of \(k\)-th powers. The authors then discuss generalizations of this process obtained by widening the choice of deletion rules. A different generalization was earlier discussed by C. T. Long [Am. Math. Mon. 73, 846–851 (1966; Zbl 0148.02003)]. Reviewer: Ian Anderson (Glasgow) MSC: 11B75 Other combinatorial number theory Keywords:combinatorial proof; Moessner’s process; squares; sequence of higher powers Citations:Zbl 0148.02003 PDFBibTeX XMLCite \textit{E. A. Bender} et al., Am. Math. Mon. 97, No. 2, 139--143 (1990; Zbl 0742.11010) Full Text: DOI