### Interesting integer sequence, or a crisis in rationalism?

#### by David Radcliffe

What is the next number in this sequence? **1, 12, 144, …**

You might think that these are powers of 12, and that the next number is 1728. But what if I told you that the sequence continues like this?

**1, 12, 144, 1750, 23420, 303240, 3641100, 46113200, 575360400, 7346545000, …**

I will explain this sequence after the jump.

This sequence was brought to my attention in tweet by M Stanley Jones, who described it as a crisis in rationalism. As a die-hard rationalist, I prefer to see it as an interesting math problem.

The numbers are successive powers of ten, written in base 8. (This is sequence A000468 in the OEIS.)

Mr. Jones takes this a step further. He reinterprets each number in the sequence as a base ten number, and then he divides each term by the previous term. At the start, each quotient is 12, but then stranger things start to happen.

This sequence has two properties which seem to be inconsistent.

- Each term is at least 12 times the previous term, and sometimes the ratio is much greater than 12.
- The
*n*th root of the*n*th term lies between 12 and 13 for all*n*> 0.

Can you prove that both of these statements hold, and explain why they do not contradict each other? What other properties of this sequence can you discover?

I like these types of deceptive number patterns. One I sometimes use in class is 1,2,4,8 with 14 rather than 16 as the next term.

Sometimes I just use the pattern on its own but it also emerges when considering the question “If n intersecting circles are drawn inside a rectangle, what is the maximum number of regions that are formed?”