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By Euler L.

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Alladi and Hoggatt, 1974). The number of distinct compositions C„ of a positive integer n in terms of 1 s and 2s is Fn+l, where n > 1. Proof. Let C„(l) and C„(2) denote the number of compositions of« that end in 1 and 2, respectively. Clearly, C,(l) = 1 andCi(2) = 0 , so C, = d ( l ) + C,(2) = 1. Likewise, C2 = C 2 (l) + C2(2) = 1 + 1 = 2 . Now consider a composition of «, where « > 3. Case 1. Suppose the composition ends in 1. Deleting the 1 at the end yields a composition of n — 1. On the other hand, adding a 1 at the end of a composition of « — 1 yields a composition of« that ends in 1.

Likewise, C2 = C 2 (l) + C2(2) = 1 + 1 = 2 . Now consider a composition of «, where « > 3. Case 1. Suppose the composition ends in 1. Deleting the 1 at the end yields a composition of n — 1. On the other hand, adding a 1 at the end of a composition of « — 1 yields a composition of« that ends in 1. Thus C„(l) = C„_i. FIBONACCI AND COMPOSITIONS WITH IS AND 2S 41 Case 2. Suppose the composition ends in 2. Deleting the 2 at the end, we get a composition of n — 2. On the other hand, by adding a 2 or two 1 s, we get a composition of n.

Using the FRR (Eq. 1), compute the value of F_„, where 1 < n < 10. 5. Using Exercise 4, predict the value of F_„ in terms of F„. 6. Compute the value of L_„, where 1 < n < 10. 7. Using Exercise 6, predict the value of L_„ in terms of L„. To commemorate the publication of the maiden issue of the Journal of Recreational Mathematics, L. Bankoff of Los Angles published his discovery that F2Q — F\g— F\5 — F$ — F\ = Fn + F\3 + F\\ + Fg + FT + Fi and that each sum gives the year. 8. Find the year in which the journal was first published.

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A solution to a problem of Fermat, on two numbers of which the sum is a square and the sum of their squares is a biquadrate, inspired by the Illustrious La Grange by Euler L.


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