Let me introduce you to a set of integers I’ve fiddled around recently.
I’ll call them
fibbons.
A fibbon is a number such that each of its digits, starting from the 3rd from the left, is a sum of 2 preceding digits e.g. 123, 3257, ..... etc
To get more profound knowledge of fibbons please fulfill the following tasks:
1. Show that the maximal length of a fibbon is 8 digits.
2. List all existing fibbons.
3. Specify the least frequent digit in your list.
4. Specify the most frequent digit in your list.
5. Denoting by f(k) the number of fibbons containing k digits, list the values for f(1), f(2), ...f(8)
6. Explain briefly how the list of 2 was generated.
(In reply to
Solution by tomarken)
The definition of a fibbon does not preclude numbers that could be extended further. Thus, not only is 7189 a fibbon, but so is 718. I'll also consider 71 and 7 to be fibbons based on task 5.
Call a "full fibbon" a fibbon that cannot be extended. You listed all of these full fibbons of 3 or more digits. There are also 45 full fibbons with 2 digits.
1 0 0
2 45 90
3 20 60
4 17 68
5 4 20
6 3 18
7 0 0
8 1 8
So by my reckoning the final column gives a grand total of 264 fibbons.
The number of k digit fibbons is the sum of all numbers k or above in the second column
1 0 90
2 45 90
3 20 45
4 17 25
5 4 8
6 3 4
7 0 1
8 1 1
This new third column gives the values for task 5.
I don't feel like counting the occurrences by digit.
I generated the a list of full fibbons by hand on paper. It only took a few minutes.
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Posted by Jer
on 2021-03-16 10:08:40 |
Extended Solution
