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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
##### Keywords:
elliptic curves; generators; finite fields
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##### References:
 [1] Kimberling, C., The Zeckendorf array equals the Wythoff array, Fibonacci Quart., 33, 3-8, (1995) · Zbl 0828.11009 [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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