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A variation on the theme of Nicomachus. (English) Zbl 1410.11079
From Nicomachus’ theorem \[1^3+2^3 + \dots + m^3 = (1+2+\ldots+m)^2\] it can be easily deduced that \[Q(\alpha,m) : =\frac{\sum_{n=1}^m \lfloor \alpha n \rfloor^3}{(\sum_{n=1}^m \lfloor \alpha n \rfloor)^2}\] has limit \(\alpha\) when \(m\) goes to infinity, \(\alpha \in \mathbb{R} \setminus \{0\}\) and \(\lfloor x \rfloor\) is the integer part.
In this short and elementary note, the authors give formula for the difference \(Q(\phi^2,m_k)-Q(\phi,m_k)\) where \(\phi=(1+\sqrt{5})/2\) and \(m_k=F_k-1\) with \(F_k\) the k-th Fibonacci number in terms of Fibonacci numbers and Lucas numbers. Further variations are proposed.

MSC:
11G20 Curves over finite and local fields
11T23 Exponential sums
11Y16 Number-theoretic algorithms; complexity
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References:
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[2] Stopple, J., A Primer of Analytic Number Theory. From Pythagoras to Riemann, (2003), Cambridge University Press: Cambridge University Press, Cambridge · Zbl 1029.11001
[3] Warnaar, S. O., ‘On the q-analogue of the sum of cubes’, Electron. J. Combin.11(1) (2004), Note 13, 2 pages. · Zbl 1071.05010
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