Exotic Arithmetic, Summer 2020 Fibonacci

We begin this essay with three tricks.1 Two of these are due to Art Benjamin. I got the third one from my new friend Derek Liu of San Diego, CA. As I type this in January 2013, Derek is six years old. Start with two numbers written one below the other. Write their sum just below the second one, and for the fourth, write the sum of the second and third. Do this until you have written 14 numbers. Now add the first 10 of these and hide your answer from me. I’ll glance at your list and instantly tell you your sum. Then compute the ratio of the 10th number to the 9th number. I can also tell you that to a high degree of accuracy. Now take the sum of first 12 numbers. I can also tell you that sum without doing the computation. How? This problem set is designed as an introduction to the famous and ubiquitous sequence of integers called the Fibonacci sequence. The sequence is

defined by letting F1 = 1 and F2 = 1, and the recurrence relation Fn = Fn−1 + Fn−2 for all n ≥ 3. Thus, F3 = 2,F4 = 3,F5 = 5. The first few terms of the sequence are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,F12 = 144. For convenience, we build a table of the first 20 values below: 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765

Let f(x) = x+x2+2x3+3x4+.... This infinite polynomial is called the generating function for the Fibonacci numbers. We can derive a closed form representation of

1Unauthorized reproduction/photocopying prohibited by law’ c

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f as follows.

∞ 2 3 X n f(x) = x + x + 2x + ... = Fnx n=1 ∞ 2 X n = x + x + Fnx n=3 ∞ 2 X n+2 = x + x + Fn+2x n=1 ∞ 2 X n+2 = x + x + (Fn + Fn+1)x n=1 ∞ ∞ 2 2 X n X n+1 = x + x + x Fnx + x Fn+1x n=1 n=1 = x + x2 + x2f(x) + x(f(x) − x) = x + x2f(x) + xf(x)

Then f(x) − xf(x) − x2f(x) = x and it follows that x f(x) = . 1 − x − x2 Our problems are essentially of two types, divisor problems and addition prob- lems. Each problem requires both experimentation and proof. The proofs typically use the Principle of Mathematical Induction (PMI). PMI says that if a proposition P about the nonnegative integers is true about 0 (or in some cases 1) and it is also true about the successor of each integers for which it is true, then it is true about all the positive integers. We’ll provide a few sample proofs as we go. Incidentally, there is another common sequence of numbers defined similarly, called the Lucas

Numbers, Ln about which we can compose similar problems. First L1 = 1 and L2 = 3. Also, Ln = Ln−1 + Ln−2 for all n ≥ 3.

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1. A monomino is a unit square (), and a domino () is a pair of monominoes glued together along a common edge. How many ways is there to fill up a 1×7 rectangle with monominoes and dominoes?

2. Experiment with the Fibonacci sequence to find a formula for the sum Sk of the first k Fibonacci numbers. Symbolically,

Sk = F1 + F2 + ··· + Fk.

This sequence begins S1 = 1,S2 = 2,S3 = 4,S4 = 7,S5 = 12. Find a pattern.

3. Find the units digit of the Fibonacci number F2017.

4. How many of the first Fibonacci numbers F1,F2,...,F1000 are multiples of 9? 5. What is the sum of the first 1000 even Fibonacci numbers? For example, the

sum of the first 2 Fibonacci numbers is F3 +F6 = 2+8 = 10. You may express your answer in terms of other Fibonacci numbers.

6. What is the sum of the first 100 evenly subscripted Fibonacci numbers? For

example, the sum of the first 4 evenly subscripted Fibonacci numbers is F2 + F4 + F6 + F8 = 1 + 3 + 8 + 21 = 33. You may express your answer in terms of other Fibonacci numbers. √ √ 1+ 5 1− 5 7. Binet’s formula. Let α = 2 and β = 2 . Then √ n n Fn = (α − β )/ 5

for all n ≥ 1. Use this formula to verify that F10 = 55 and F20 = 6765.

8. Use Binet’s formula to find the number of digits in F2013.

9. The first Fibonacci number that is divisible by 11 is F10 = 55. Find the second and the third. Develop a conjecture about which Fibonacci numbers are multiples of 11 and prove it.

10. In this problem, and in the one that follows, we start with G1 = a and G2 = b, where a and b are positive integers. We define a sequence just as above by the

recurrence Gn+2 = Gn + Gn+1. Now define yet a third sequence Hn as follows: th Hn = Gn + Fn, where Fn denotes the n Fibonacci number. Prove that the new sequence Hn also satisfies the recurrence relation Hn+2 = Hn +Hn+1. Let the sequence Gn be defined as above with a = 2. Suppose G15 = 843, What is G16?

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∞ X Fi 11. Consider the series . Does it converge? If so, find its sum. 10i i=1

12. Let Tn denote the number of ways to tile a 2 × n board with dominos. Find T10,T11, and T12.

13. Let Zn denote the number of ways to tile a 3 × n board with dominos and one monomino. Find Z3,Z4, and Z5.

14. Let S be a subset of the set {1, 2, 3, 4,..., 100}, and suppose that no three elements of S are the lengths of the sides of a non-degenerate triangle. What is the largest number of elements S can have?

15. Let K be the integer you found in the problem above. How many subsets of {1, 2, 3, 4,..., 100} with K elements satisfy the property that no three of its members are the lengths of the sides of a non-degenerate triangle?

16. How many of the first 2006 Fibonacci numbers are multiples of

(a) 99 (b) 66 (c) 81

17. Find the number of digits in the Fibonacci number Fn.

18. Consider the Zometool triangular arrangement given below. C

•...... y ...... z ...... •...E ...... x ...... D ...... •...... 1 ...... 1 ...... 1 ...... ••...... ABy

The structure is built with blue struts of lengths x, 1, y, z, where x < 1 < y < z.

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Triangles ABD and AEC are similar, and triangles ABC, DEC, and BEA are similar.

(a) Find the lengths of y and z in terms of x. (b) Solve for x, y and z. (c) Solve x2 − x − 1 = 0 for two roots. Call the smaller on β and the larger one α. Compute each of the numbers α2, α3, α4, α5, writing each of these in terms of α itself. Then do the same for β.

19. What proportion of the Fibonacci numbers are even? What fraction of them are multiples of 3? multiples of 4? 5? Is there a pattern to these answers? I read this problem at Jim Tanton’s webpage.

20. The first four terms of a sequence are 2, 6, 12, 72. Each terms is obtained from the previous 2 my multiplying. When the ninth terms is written in the form 2a3b, what are a and b?

21. Define Fn as F1 = 1,F2 = 1,Fn+2 = Fn+1 + Fn for n ≥ 1. Compute F F F F S = 1 + 2 + 3 + 4 + ··· 5 52 53 54

F 22. Find lim n+1 . n→∞ Fn

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