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Adding up to powers. (English) Zbl 0742.11010

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)].

MSC:

11B75 Other combinatorial number theory

Citations:

Zbl 0148.02003
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