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Lemma: Sums of Floors

For any real number $\alpha\in\mathbb R$ and any two integers $x,y\in\mathbb Z$ with $y\neq 0,$ the following closed formula for the sum of floor function holds:
$$\sum_{k=0}^{y-1}\left\lfloor\frac{xk+\alpha}m\right\rfloor=d\left\lfloor\frac{\alpha}d\right\rfloor +\frac{y-1}{2}x+\frac{d-y}{2},$$
where $d=\gcd(x,y)$ is the greatest common divisor of $x$ and $y.$

| | | | | created: 2019-06-02 08:20:59 | modified: 2019-06-02 08:38:54 | by: bookofproofs | references: [1112]

1.Proof: (related to "Sums of Floors")


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Bibliography (further reading)

[1112] Graham L. Ronald, Knuth E. Donald, Patashnik Oren: “Concrete Mathematics”, Addison-Wesley, 1994, 2nd Edition

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