Numbers

This is an update of an earlier post. The formula is slightly different and the results are even more remarkable.

I like discovering and exploring mathematical sequences. There is one that, like the Fibonacci sequence, uses the sum of the previous two terms to find the next term. However, a multiple of n is added. Then certain factors are removed from the sum where these are found before it is accepted as the next term.

a and b are the two initial terms to get things going.

n > 2 ∴       tn    =    (tn-1 + tn-2 + dn )   ÷   (  GCD of   c  and   (tn-1 + tn-2 + dn)  )
Where c is the product of the factors being removed, so if you are removing one factor of 2 and one factor of 3 (where these occur), for instance, then c = 6.

d is any positive integer.

These sequences increase remarkably slowly, on average, but do so with terms of fluctuating size. For one example out of many, where a = 1, b = 1, c = 6, and d =1, the 200,000th term is only a 6-digit number. None of the first 26,000 terms have more than 13 digits, so this could well apply to the first 200,000 terms as a whole. It certainly doesn’t look as if any of the terms will have many more than 13 digits. This is striking enough when d = 1, but when I increased d to a value of two, three, four, five digits… the sequence behaves in the same way.

What makes this so striking is that if you remove the dn part of the formula (or, in other words: d = 0), these sequences usually grow very rapidly and presumably indefinitely, or they run into an unending cycle of small, usually single-digit, terms, as you would expect if chance is all that’s involved. Nevertheless, it can still take thousands of terms before this happens – a balance of ‘forces’ that aroused my curiosity. Increasing d from a zero to a positive integer turned out to be the magic ingredient required. The dn part of the formula suggests that the terms will increase on average as n increases but, as long as d is greater than zero, they never ‘collapse into a loop’. And they never ‘run out of control’ as the Fibonacci sequence does.

It seems that chance governs which terms appear but, given the stability of these sequences, something else is clearly going on; something is stabilizing them. The 200,000th Fibonacci number has, off the top of my head, between 40,000 and 50,000 digits – compare this with the number of atoms in the universe being an 85-digit number (and it took ages to count, I must say!) These sequences exactly resist this tendency to expand, staying remarkably close to small values by comparison; yet they do not ‘collapse into a loop’.

I would be interested to hear if someone has an explanation for this.The tendency of Fibonacci-like sequences (where a and b can be any integers) to increase