Summing discrete 3d coordinate lengths

Question

Consider a three dimensional discrete coordinate system $(x,y,z)$, where $x,y,z\in$ natural numbers.

The number of digits describing an integer coordinate for each dimension is $l_c=\lfloor log(c) \rfloor+1$, where $c$ is $x$, $y$ or $z$.

The total number of digits describing a point in the space is $l=\lfloor log(x) \rfloor+\lfloor log(y) \rfloor+\lfloor log(z) \rfloor+3$.

I am looking for a formula to describe how many points each total coordinate digit length can describe.

Any suggestions?

Example:

As one digit for each dimension can describe a total of $(10)(10)(10)$ points, the total coordinate length $3$ can describe a total of $10^3$ points in the space.

Two digits for any of the dimensions and one digit for the rest gives a total number of points possible to describe as $(3)(10^2-10)(10)(10)$. In other words, a total of four digits can describe a maximum of $27000$ points in the space.

A total of five coordinate digits can describe a maximum of $$(3)(10^3-10^2)(10)(10)+(3)(10^2-10)^2(10)=513000$$ points in the space.

Six digits can describe a total number of $$(3)(10^4-10^3)(10)(10)+(6)(10^3-10^2)(10^2-10)(10)+(10^2-10)^3=8289000$$ points in the space.

And so on.

Any suggestions on how to produce a formula are greatly appreciated.

Answer 1

OPs approach is fine and in fact the examples already present all possible different variants of the number of digits $k_1,k_2,k_3$ of the three dimensions with $k_1+k_2+k_3=n$ digits.

• All three dimensions are equal: $\qquad\qquad\ k_1=k_2=k_3$

• Two are equal, the third is different: $\qquad\ k_1=k_2, k_1\ne k_3$

• All three are pairwise different: $\qquad\qquad\; k_1\ne k_2, k_1\ne k_3, k_2\ne k_3$

As there are some different cases to distinguish we conveniently use Iverson brackets.

Let $n\ge 3$. We start with \begin{align*} \sum_{{1\leq k_1\leq k_2\leq k_3\leq n}\atop{k_1+k_2+k_3=n}} &\left\{\left(10^{k_1}-10^{k_1-1}[[k_1>1]]\right)^3[[k_1=k_2=k_3]]\right.\tag{1}\\ &\qquad+3\left(10^{k_1}-10^{k_1-1}[[k_1>1]]\right)^2\left(10^{k_3}-10^{k_3-1}[[k_3>1]]\right)\\ &\qquad\qquad\cdot[[k_1=k_2,k_1\ne k_3]]\\ &\qquad+3\left(10^{k_1}-10^{k_1-1}[[k_1>1]]\right)\left(10^{k_3}-10^{k_3-1}[[k_3>1]]\right)^2\tag{2}\\ &\qquad\qquad\cdot[[k_1\ne k_2,k_2= k_3]]\\ &\qquad\left.+6\prod_{j=1}^3\left(10^{k_j}-10^{k_j-1}[[k_j>1]]\right)[[k_1\ne k_2,k_1\ne k_3,k_2\ne k_3]]\right\}\tag{3} \end{align*}

We can simplify the three summands (1), (2) and (3) somewhat.

Case 1: All three dimensions are equal

We obtain \begin{align*} \sum_{{1\leq k_1\leq k_2\leq k_3\leq n}\atop{k_1+k_2+k_3=n}}&\left(10^{k_1}-10^{k_1-1}[[k_1>1]]\right)^3[[k_1=k_2=k_3]]\\ &=\sum_{{1\leq k_1= k_2= k_3\leq n}\atop{k_1+k_2+k_3=n}}\left(10^{k_1}-10^{k_1-1}[[k_1>1]]\right)^3\\ &\,\,\color{blue}{=\left(10^{\frac{n}{3}}-10^{\frac{n}{3}-1}[[n>3]]\right)^3[[3|n]]}\tag{4} \end{align*}

Since $k_1=k_2=k_3$ we have only one case to consider, namely $3k_1=n$ resp. $k_1=\left\lfloor\frac{n}{3}\right\rfloor$. This implies that $n$ has to be a multiple of $3$ which is asserted by $[[3|n]]$, otherwise the sum is zero.

Case 2: Two are equal, the third is different

We obtain \begin{align*} &3\sum_{{1\leq k_1\leq k_2\leq k_3\leq n}\atop{k_1+k_2+k_3=n}} \left(10^{k_1}-10^{k_1-1}[[k_1>1]]\right)^2\left(10^{k_3}-10^{k_3-1}[[k_3>1]]\right)\\ &\qquad\qquad\cdot[[k_1=k_2,k_1\ne k_3]]\\ &\qquad+3\sum_{{1\leq k_1\leq k_2\leq k_3\leq n}\atop{k_1+k_2+k_3=n}} \left(10^{k_1}-10^{k_1-1}[[k_1>1]]\right)\left(10^{k_3}-10^{k_3-1}[[k_3>1]]\right)^2\\ &\qquad\qquad\qquad\cdot[[k_1\ne k_2,k_2= k_3]]\\ &\quad=3\sum_{{1\leq k_1= k_2< k_3\leq n}\atop{2k_1+k_3=n}} \left(10^{k_1}-10^{k_1-1}[[k_1>1]]\right)^2\left(10^{k_3}-10^{k_3-1}[[k_3>1]]\right)\\ &\qquad+3\sum_{{1\leq k_1< k_2= k_3\leq n}\atop{k_1+2k_3=n}} \left(10^{k_1}-10^{k_1-1}[[k_1>1]]\right)\left(10^{k_3}-10^{k_3-1}[[k_3>1]]\right)^2\\ &\quad\,\,\color{blue}{=3\sum_{k_1=1}^{\left\lfloor\frac{n-1}{3}\right\rfloor} \left(10^{k_1}-10^{k_1-1}[[k_1>1]]\right)^2\left(10^{n-2k_1}-10^{n-2k_1-1}\right)}\\ &\qquad\,\,\color{blue}{+3\sum_{k_3=\left\lceil\frac{n+1}{3}\right\rceil}^{\left\lfloor\frac{n-1}{2}\right\rfloor} \left(10^{n-2k_3}-10^{n-2k_3-1}[[n-2k_3>1]]\right)\left(10^{k_3}-10^{k_3-1}\right)^2}\tag{5} \end{align*}

The upper limit $\left\lfloor\frac{n-1}{3}\right\rfloor$ of the left-hand sum in (5) follows from the index region \begin{align*} &1\leq k_1=k_2<k_3\leq n\\ &2k_1+k_3=n\qquad\qquad\qquad\to\qquad k_3=n-2k_1>k_1\qquad\to\qquad k_1<\frac{n}{3}\\ \end{align*}

The lower limit $\left\lceil\frac{n+1}{3}\right\rceil$ of the right-hand sum in (5) follows from the index region \begin{align*} &1\leq k_1<k_2=k_3\leq n\\ &k_1+2k_3=n\qquad\qquad\qquad\to\qquad k_1=n-2k_3<k_3\qquad\to\qquad k_3>\frac{n}{3}\\ \end{align*}

The upper limit $\left\lfloor\frac{n-1}{2}\right\rfloor$ of the right-hand sum in (5) follows from the index region \begin{align*} &1\leq k_1<k_2=k_3\leq n\\ &k_1+2k_3=n\qquad\qquad\qquad\to\qquad n-2k_3\geq 1\qquad\to\qquad k_3\leq \frac{n-1}{2}\\ \end{align*}

Case 3: All three are pairwise different

We obtain \begin{align*} &6\sum_{{1\leq k_1\leq k_2\leq k_3\leq n}\atop{k_1+k_2+k_3=n}} \prod_{j=1}^3\left(10^{k_j}-10^{k_j-1}[[k_j>1]]\right)[[k_1\ne k_2,k_1\ne k_3,k_2\ne k_3]]\\ &\qquad=6\sum_{{1\leq k_1< k_2< k_3\leq n}\atop{k_1+k_2+k_3=n}} \prod_{j=1}^3\left(10^{k_j}-10^{k_j-1}[[k_j>1]]\right)\\ &\qquad\,\,\color{blue}{=6\sum_{k_1=1}^{n-2}\sum_{k_2=k_1+1}^{\left\lfloor\frac{n-k_1-1}{2}\right\rfloor}\left(10^{k_1}-10^{k_1-1}[[k_1>1]]\right)\left(10^{k_2}-10^{k_2-1}\right)}\\ &\qquad\qquad\qquad\qquad\qquad\color{blue}{\cdot\left(10^{n-k_1-k_2}-10^{n-k_1-k_2-1}\right)}\tag{6} \end{align*}

Putting (4) - (6) together gives a simplified formula of (1) - (3).

The upper limit $\left\lfloor\frac{n-k_1-1}{2}\right\rfloor$ of the sum in (6) follows from the index region \begin{align*} &1\leq k_1< k_2< k_3\leq n\\ &k_1+k_2+k_3=n\qquad\qquad\to\qquad k_3=n-k_1-k_2>k_2\qquad\to\qquad k_2<\frac{n-k_1}{2}\\ \end{align*}