## Fibonacci and Lucas Numbers and the Golden Section: Theory by Steven Vajda

By Steven Vajda

The Fibonacci precept of forming a brand new quantity via a suitable mix of prior numbers has been prolonged to yield sequences with marvelous and infrequently mystifying homes: the Meta-Fibonacci sequences. this article examines Meta-Fibonacci numbers, continuing to a survey of the Golden part within the aircraft and area. It additionally describes Platonic solids and a few in their much less regular gains, and an appendix and different supplementations provide beneficial heritage details. scholars and lecturers will locate this publication appropriate to stories of algebra, geometry, chance concept, computational features, and combinatorial elements of quantity theory.

Steven Vajda used to be born in Budapest in 1901 and died in England in 1995. For the last twenty-two years of his existence, he was once traveling Professor of arithmetic at Sussex University. As a trendy instructor, lecturer, and writer he performed a necessary position within the improvement of mathematical programming and operations examine and wrote greater than a dozen books and plenty of examine papers on those and different subject matters together with video game theory.

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**Example text**

Now set n = 0, h = k. We derive from (20b) (22) It follows from (22) and (17c) that L 2h - 2( - l)h =SF~ (23) and from (23) and (17c) that (24) 5. We concentrate now on formula (24), a special case of the so-called Pell's equation. (Historians tell us that this is a misnomer, and the study of this type of Diophantine equation should more correctly be attributed to Fermat. ) (24) shows that F1i and Lh cannot have any common divisor larger than 2, and that either both Fh and L1i are even, or both are odd.

IV(b). n 2: G;+2G;-1 = G~+1 - Gf. (43) i= I 10. We have n n n n i= I i= I i= I i= 1 2: Gf= 2: G;(G;+ 1-G;_ 1)= 2: G;G;+ 1- 2: G;- 1 G; that is n 2: Gf = GnGn+1 -GoG1. i= 1 (44) Relationships 44 If G; [Ch. III =F;, then n L Ff= FnFn+l· (45) i= 1 This formula can also be illustrated geometrically. We do it in Fig. V(a). (See 13X21=F7 xF8 p2 7 p2 4 52 Fig. V(a). Ch. ) As n increases, the rectangle tends to a Gold Rectangle, because Jim n= F~+i tends oo n to 't (see (101) in Chapter VIII). In Fig. V(a) we have also drawn the straight lines which connect the centres C; of alternate squares.

1, hence a = 1, P= 1. Thus (57) p= -11\15. \15 and For the (58) (59) Formula (58) was known to De Moivre (1718) and rediscovered by Binet (1843), and also by Lame (1844) after whom the sequence is sometimes called. Fibonacci numbers as well as Lucas numbers are integers. tn and an are, of course, also generalized Fibonacci numbers, though integers only for n = 0. Nevertheless, formula (8) applies to them as well. For instance, with n = 0, From (58) and (59) it follows, trivially, that tn = l(Ln + \15Fn) and From (56) we notice that a and a= m n\15 2+10 an = i(L 11 - \15Fn).