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2016e.00849 Treeby, David Tang, Jenny Combinatorial derivations of familiar identities. Parabola 51, No. 1, 3 p., electronic only (2015). 2015 AMT Publishing, Australian Mathematics Trust, University of Canberra, Canberra; School of Mathematics \& Statistics, University of New South Wales, Sydney EN K20 I30 combinatorics enumeration problems set of consecutive integers enumerating rising sequences sums of squares
• https://www.parabola.unsw.edu.au/files/articles/2010-2019/volume-51-2015/issue-1/vol51_no1_2.pdf
• From the text: Finding two ways to enumerate the same collection of objects can often give rise to useful formulae. For instance, the sum $1+2+\dots +n$ can be interpreted as the maximum number of different handshakes between $n+1$ people. The first person may shake hands with $n$ other people. The next person may shake hands with $n-1$ other people, not counting the first person again. Continuing like this gives the above sum. Another approach is to simply realise that each of the $n+1$ guests shakes hands with $n$ other guests. However, this counts handshakes twice. Therefore, $1+2+\dots +n=\frac{n(n+1)}{2}$. This article concerns a combinatorial argument that gives rise to the familiar formula for the sum of the first $n$ squares, $1^2+2^2+\dots +n^2$. For our derivation of the formula we enumerate the same collection of objects two different ways and then equate the results.