While playing around with the patterns produced by some sequences, I discovered some interesting patterns. It began with the sequence produced by
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Figure 1 |
What's interesting is the hexagonal arrangement of the points. The values cannot exceed 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: for values of from 0 to 4000. See Figure 2 (permalink).
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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
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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.
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Figure 4 |
As can be seen from Figure 4, negative value of 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: where the two bounding lines are given by and .
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