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