Visualising Sequences

While playing around with the patterns produced by some sequences, I discovered some interesting patterns. It began with the sequence produced by

$$\text{a}(n)=\sin(n) \cdot e^{-0.001 \cdot n}$$ for values of $n$ from 0 to 2000. See Figure 1 (permalink).

Figure 1

What's interesting is the hexagonal arrangement of the points. The values cannot exceed $\pm$1 and the exponential component brings progressive values closer and closer to zero, although at a very slow rate. The pattern becomes rather different once we introduce another element as follows: $$\text{a}(n)=n \cdot \sin(n) \cdot e^{-0.001 \cdot n}$$ for values of $9n$ from 0 to 4000. See Figure 2 (permalink).

Figure 2

Values range from about - 367 to + 367. Once again, the exponential component eventually dominates and the range of values inexorably decrease. Increasing the exponent of the $n$ element doesn't really alter the pattern. For example, raising the $n$ to the fourth power:

$$\text{a}(n)=n^4 \cdot \sin(n) \cdot e^{-0.001 \cdot n}$$ for values of $n$ from 0 to 8000 produces the pattern shown in Figure 3 (permalink).

Figure 4

So nothing profound in this post, just interesting now inputting integral values into a function and plotting the output produces interesting patterns. You can see the same general shape using a program like GeoGebra but the patterns shown do not emerge. See Figure 4.

Figure 4

As can be seen from Figure 4, negative value of $n$ cause the output values to explode. Of course, most of the sequences that I examine in this blog are integer sequences, the result of integer output from integer input, as found in the OEIS. It's interesting however, from time to time, to examine non-integer output from integer input, as I've done in this post. 

Finally, before we do, consider Figure 5 (permalink) that shows an interesting result for a sequence generated by: $$\text{a}(n)=n \cdot e^{ \, \sin(x)}$$ where the two bounding lines are given by $y=e \cdot x$ and $y=1/e \cdot x$.

Figure 5