There Can Be Only One 2

Let's recall the Fibonacci-like sequence from my previous blog post titled There Can Be Only One 1.

We start with a digit \(d\) between 0 and 9 and any integer \(s\) that does not contain \(d\). For example, 1 and 2 would be suitable. After that we begin the Fibonacci process using \(d\) and \(s\) as our seeds except that whenever a number appears containing one or more digits \(d\), they are all removed. 

In my previous post, I looked at three scenarios:

• \(d=1\) and \(s=2\)
• \(d=1\) and \(s=24\)
• \(d=1\) and \(s=70\)

A major change occurs when we consider \(d=2\) as our "forbidden digit". Suddenly the 100 barrier can be breached as can be seen in Figure 1 where the sequence for \(d=2\) and \(s=1\) is depicted. Here the resulting sequence does not loop but exceeds the upper bound of 40000 that was set.

Figure 1: permalink

The sequence progresses like this:

[2, 1, 3, 4, 7, 11, 18, 9, 7, 16, 3, 19, 0, 19, 19, 38, 57, 95, 15, 110, 15, 15, 30, 45, 75, 10, 85, 95, 180, 75, 55, 130, 185, 315, 500, 815, 1315, 130, 1445, 1575, 300, 1875, 175, 50, 5, 55, 60, 115, 175, 90, 65, 155, 0, 155, 155, 310, ...

We can see that 155 + 155 = 310 jumps over the 200 barrier and from there it becomes seemingly unbounded (I tested up to ten million).

Not all values of \(s\) lead to unbounded sequences. For example, \(s=5\) leads quickly to a loop (see Figure 2): 

Starting Parameters: Forbidden Digit = 2, Second Number = 5

Max Value Limit: 40000

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Resulting Sequence (17 terms):

[2, 5, 7, 1, 8, 9, 17, 6, 3, 9, 1, 10, 11, 1, 1, 0, 1]

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Conclusion: The sequence entered a continuous loop.

The repeating pair that triggered the stop was: (0, 1)

Figure 2: permalink

Sequences are unbounded for \(s=1, 3, 4\) but loop for \(s=5, 6, 7, 8, 9, 10\) before becoming unbounded again for \(s=11\).