Fibonacci numbers

Article

October 25, 2021

Fibonacci numbers (pronounced fibonaccsi) are elements of one of the best known second-order recursive series in mathematics. The zero element is 0, the first element is 1, and the further elements are obtained as the sum of the previous two. Formula: F n { 0 , if n 0 ; 1 , if n 1 ; F n - 1 + F n - 2 , if n ≥ 2. {\ displaystyle F_ {n} {\ begin {cases} 0, & {\ mbox {ha}} n 0; \\ 1, & {\ mbox {ha}} n 1; \\ F_ {n- 1} + F_ {n-2}, & {\ mbox {ha}} n \ geq 2. \ end {cases}}} Fibonacci numbers form an infinite, ascending series; the first few elements of this, starting from the zero, are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34. Several large lists of Fibonacci numbers can be freely downloaded from the Internet.

Origin

The series was first described in 1150 by two Indian mathematicians, Gopala and Hemacsandra, who, when studying the theoretical issues of Sanskrit poetry, encountered a problem of decomposition into a sum (how many short syllables can be filled in if a long syllable equals two?) . In the West, he found it independently in 1202 by Fibonacci, who in Liber Abaci (Book on the Abacus) gave up the growth of an imaginary rabbit family as a practice task: how many pairs of rabbits will be in n months if we assume that there is only one newborn pair of rabbits in the first month; newborn rabbit pairs become fertile within two months; each fertile pair of rabbits gives birth to another pair each month; and do rabbits live forever? ”Kepler rediscovered in his book De nive sexangula (The Hexagonal Snowflake) in 1611 and associated it with various natural phenomena. The name used today came from E. Lucas.

Binet formula

Proportion of adjacent Fibonacci numbers ( F n + 1 / F n {\ displaystyle F_ {n + 1} / F_ ​​{n} \,} ) ϕ {\ displaystyle \ phi \,} to the value of the golden ratio: say x 2 x + 1 {\ displaystyle x ^ {2} x + 1 \,} , and for this quadratic equation just ϕ {\ displaystyle \ phi \,} and 1 - ϕ {\ displaysty

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