**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")*

(none)

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

FeedsAcknowledgmentsTerms of UsePrivacy PolicyImprint

© 2018 Powered by BooOfProofs, All rights reserved.

© 2018 Powered by BooOfProofs, All rights reserved.