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.
Let a,b bet the first two digits of the number, with 1 <= a <= 9 and 0 <= b <= 9. Then the 3rd digit is equal to (a+b).
The 4th digit is equal to b + (a+b) = a + 2b.
The 5th digit is equal to 2a + 3b.
...
The 9th digit would be equal to 13a + 21b, but there is no choice of a that would make this result a single digit. So therefore there cannot be a 9-digit fibbon, and 8 is the maximal length.
To generate a list of fibbons, start with any such pair of digits a,b whose sum does not exceed 9 (this assumes that a fibbon by definition must be at least three digits long).
For a = 1 there are 9 choices for b (0 through 8).
For a = 2 there are 8 choices for b (0 through 7).
...
For a = 9 there is one choice for b (0). Summing those up there are exactly 45 possible fibbons.
The fibbon of maximal length is the one beginning with a,b = 1,0 which results in 10112358.
The least frequent digit is 0, which can only ever appear as the 2nd digit of a fibbon and therefore only appears 9 times in the list.
The most frequent digits are 1,2,3,5, and 8 which each appear 19 times in the list.
f(3) = 20
f(4) = 17
f(5) = 4
f(6) = 3
f(7) = 0
f(8) = 1
The full list:
10112358
202246
112358
303369
21347
40448
12358
31459
5055
2246
4156
6066
1347
3257
5167
7077
2358
4268
6178
8088
1459
3369
5279
7189
9099
246
156
437
347
257
167
628
538
448
358
268
178
819
729
639
549
459
369
279
189
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Posted by tomarken
on 2021-03-16 07:47:36 |
Solution
