A paucity of information about a number always challenges me to find something interesting about the number. Such was the case with the number associated with my diurnal age today: \( \textbf{27796} \). I could see that the first two digits of the number, 2 and 7, added to give the fourth digit and the fourth digit plus the fifth digit gave the rightmost digit (6) of the resultant sum (16). This was almost a Fibonacci sequence mod 10 except that the repeated digit 7 got in the way.
So I devised the following set of rules.
- Let \(a\) and \(b\) be the first two digits of the sequence \( (a,b)\)
- If \(b \neq 7 \), \(c=a+b \pmod {10}\) gives the next digit
- Let \(a,b=b,c\)
- If \(b=7\) then \(c=7\) and \(d=a +b \pmod {10} \) give the next two digits
- Let \(a,b=c,d\)
Applying these rules when \(a=2\) and \(b=7\) gives the following 68 member sequence (permalink):
[2, 7, 7, 9, 6, 5, 1, 6, 7, 7, 3, 0, 3, 3, 6, 9, 5, 4, 9, 3, 2, 5, 7, 7, 2, 9, 1, 0, 1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 7, 0, 7, 7, 7, 7, 4, 1, 5, 6, 1, 7, 7, 8, 5, 3, 8, 1, 9, 0, 9, 9, 8, 7, 7, 5]
How might we define 27796 in terms of this sequence? Well, perhaps like this. 27796 is the concatenation of the first five digits of the previous sequence created according to the rules 1 to 5 defined above. The digits of the sequence repeat when the 69th term is reached. The final two digits, 7 and 5, mark the end of the sequence because:$$ \begin{align} 7 + 5 &= 2 \pmod {10} \\ 5 + 2 &= 7 \end{align} $$Thus we have the same two digits, 2 and 7, that we started with. Figure 1 shows the progression of digits.
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Figure 1: permalink |
It can be considered that we are building a 68 digit number from a starting two digit number of 27. We might represent the number thus (with the superscript 7 representing the digit that is repeated and the overline indicating the cycle of 68 digits):$$ ^7 \, \overline{27796 \dots 98775}$$The number builds as follows:$$27, 277, 2779, 27796, \dots$$If we start with \(a=0\) and \(b=1\), as in the classic Fibonacci sequence, we still end up with a cycle of 68 and 7 is still the digit that's repeated. Here is the sequence (permalink) with Figure 2 showing the progression in graphical format:
[0, 1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 7, 0, 7, 7, 7, 7, 4, 1, 5, 6, 1, 7, 7, 8, 5, 3, 8, 1, 9, 0, 9, 9, 8, 7, 7, 5, 2, 7, 7, 9, 6, 5, 1, 6, 7, 7, 3, 0, 3, 3, 6, 9, 5, 4, 9, 3, 2, 5, 7, 7, 2, 9, 1, 0]
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| Figure 2: permalink |
It's easy to modify the algorithm so that a digit other than 7 gets repeated. Let's repeat the digit 6. In this case we end up with following 64 member looping sequence (permalink) with Figure 3 showing the progression in graphical format:
[2, 7, 9, 6, 6, 5, 1, 6, 6, 7, 3, 0, 3, 3, 6, 6, 9, 5, 4, 9, 3, 2, 5, 7, 2, 9, 1, 0, 1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 0, 7, 7, 4, 1, 5, 6, 6, 1, 7, 8, 5, 3, 8, 1, 9, 0, 9, 9, 8, 7, 5]
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| Figure 3: permalink |
This time we can consider that we are building a 64 digit number:$$ ^6 \, \overline{27966 \dots 99875}$$The number builds as follows:$$27, 279, 2796, 27966, \dots$$Thus 27966 could similarly be described as the concatenation of the first five digits of the previous sequence created according to the rules 1 to 5 defined above. The digits of the sequence repeat when the 65th term is reached. There are many other combinations of digits that could be explored but the point of this post is that interesting number patterns can always be found with just a little scrutiny and imagination.
I've made two previous posts about numbers whose digits display Fibonacci-like properties. These posts are Additive Fibonacci-like Numbers in August 2024 and Consolidating Fibonacci-like Numbers in November of 2024. In the former post I looked at numbers like 28191 where the arithmetic digital root was invoked to reveal a Fibonacci-like progression:$$ \begin{align} 2+8 &=10 \rightarrow 1 \\8+1 &= 9 \\1+9 &=10 \rightarrow 1 \end{align}$$Here the digital root of the two previous digits determines the next digit which is then concatenated with the previous digits. This process again produces a looping sequence of digits and a 24 digit number that repeats endlessly (permalink):$$ \begin {align} &2, 8, 1, 9, 1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, 4, 1, 5, 6\\\ &\overline{281911235843718988764156} \end{align}$$Alternatively, the digital root of all the previous digits can be used to determine the next digit. For example, 26875 where:$$ \begin{align} 2 + 6 &= 8\\ 2+6 + 8 &=16 \rightarrow 7\\ 2 + 6 + 8 + 7 &=23 \rightarrow 5 \end{align}$$In Consolidating Fibonacci-like Numbers, I considered numbers in base 10 that display Fibonacci-like properties when converted to other bases. I did not use digital roots for these cases, only digit sums. For example, 27802 in base 14:$$ \begin{align} 27802 &=\text{a1bc}_{14}\\ \text{a+1}&=\text{b} \\ \text{1+b} &=\text{c} \end{align}$$Related: The Pisano Period.
