This is very likely not what you had in mind, but mathematically there are always ways to find the "next" number in a sequence.

The simplest is finding the differences between numbers, then the difference between those differences, etc, until you get to all zeros or a single number. Once you get to a single number, you declare all higher differences to be 0. This is the same as saying you consider the original sequence the output of a polynomial resulting from fixed increases in X. The method uses the lowest-order polynomial to fit all the points, then extrapolates new values from there.

Analysis applied to your sequence:

0    4    8   21   52   65   96
  4    4   13   31   13   31
     0    9   18  -18   18
       9    9  -36   36
          0  -45   72
          -45  117
             162

In this case we had to go all the way to a single number since none of the rows was all-zero. Of course there are an infinite number of possible answers. We pick the one that is most "simple" by declaring all higher deltas to be 0. Applied to your sequence, to find the next number we start at the bottom and compute the next row on the right:

0    4    8   21   52   65   96  532
  4    4   13   31   13   31  436
     0    9   18  -18   18  405
       9    9  -36   36  387
          0  -45   72  351
          -45  117  279
             162  162

The answer therefore is 532. Note that we have actually created a continuous function that returns the input values exactly, but that allows us to find the value anywhere. There are an infinite number of functions that can do that, so we added the additional constraint that it be a polynomial with the minimum necessary order.

Again, I understand this is very unlikely what you had in mind. However, it is as valid as any solution given your under-constrained problem.

posted 7 months ago

CC BY-SA 4.0

7mo ago

Olin Lathrop‭

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