Suppose that $r$ is an irrational number with fractional part between $1/3$ and $2/3$. Let $D_n$ be the number of distinct $n$th differences of the sequence $(\lfloor{kr}\rfloor)$. It appears that

$$D_n=(2,3,3,5,4,7,5,9,6,11,7,13,8,\ldots),$$

essentially A029579. Can someone verify that what appears here is actually true?

Example: for $r=(1+\sqrt{5})/2$, we find

$$(kr)=(1,3,4,6,8,9,11,12,14,16,17,\ldots) = A000201 = \text{ lower Wythoff sequence,}$$

\begin{align*} \Delta^1 =&(2,1,2,2,1,2,1,2,2,1,2,2,1,\ldots), D_1=2 \\ \Delta^2 =&(1,-1,1,0,-1,1,-1,1,0,-1,1,\ldots), D_2=3 \\ \Delta^3 =&(-2,2,-1,-1,2,-2,2,-1,-1,2,\ldots), D_2=3. \end{align*}

Suppose that $r$ is an irrational number with fractional part between $1/3$ and $2/3$. Let $D_n$ be the number of distinct $n$th differences of the sequence $(\lfloor{kr}\rfloor)$. It appears that

$$D_n=(2,3,3,5,4,7,5,9,6,11,7,13,8,\ldots),$$

essentially A029579. Can someone verify that what appears here is actually true?

Example: for $r=(1+\sqrt{5})/2$, we find

$$(\lfloor{kr}\rfloor)=(1,3,4,6,8,9,11,12,14,16,17,\ldots) = A000201 = \text{ lower Wythoff sequence,}$$

\begin{align*} \Delta^1 =&(2,1,2,2,1,2,1,2,2,1,2,2,1,\ldots), & D_1=2, \\ \Delta^2 =&(1,-1,1,0,-1,1,-1,1,0,-1,1,\ldots), & D_2=3, \\ \Delta^3 =&(-2,2,-1,-1,2,-2,2,-1,-1,2,\ldots), & D_2=3. \end{align*}

Suppose that $r$ is an irrational number with fractional part between $1/3$ and $2/3$. Let $D_n$ be the number of distinct $n$th differences of the sequence $(\lfloor{kr}\rfloor)$. It appears that

$$D_n=(2,3,3,5,4,7,5,9,6,11,7,13,8,\ldots),$$

essentially A099579. Can someone verify that what appears here is actually true?

Example: for $r=(1+\sqrt{5})/2$, we find

$$(kr)=(1,3,4,6,8,9,11,12,14,16,17,\ldots) = A000201 = \text{ lower Wythoff sequence,}$$

\begin{align*} \Delta^1 =&(2,1,2,2,1,2,1,2,2,1,2,2,1,\ldots), D_1=2 \\ \Delta^2 =&(1,-1,1,0,-1,1,-1,1,0,-1,1,\ldots), D_2=3 \\ \Delta^3 =&(-2,2,-1,-1,2,-2,2,-1,-1,2,\ldots), D_2=3. \end{align*}

Suppose that $r$ is an irrational number with fractional part between $1/3$ and $2/3$. Let $D_n$ be the number of distinct $n$th differences of the sequence $(\lfloor{kr}\rfloor)$. It appears that

$$D_n=(2,3,3,5,4,7,5,9,6,11,7,13,8,\ldots),$$

essentially A029579. Can someone verify that what appears here is actually true?

Example: for $r=(1+\sqrt{5})/2$, we find

$$(kr)=(1,3,4,6,8,9,11,12,14,16,17,\ldots) = A000201 = \text{ lower Wythoff sequence,}$$

\begin{align*} \Delta^1 =&(2,1,2,2,1,2,1,2,2,1,2,2,1,\ldots), D_1=2 \\ \Delta^2 =&(1,-1,1,0,-1,1,-1,1,0,-1,1,\ldots), D_2=3 \\ \Delta^3 =&(-2,2,-1,-1,2,-2,2,-1,-1,2,\ldots), D_2=3. \end{align*}