fibs

Fibonacci numbers, commonly referred to as fibs, are a sequence of integers defined by F(0)=0, F(1)=1 and F(n)=F(n-1)+F(n-2) for n≥2. The sequence begins 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on.

Origin and naming: The sequence is named after the Italian mathematician Leonardo of Pisa, known as Fibonacci, who introduced it to European mathematics in Liber Abaci (1202). The numbers had appeared earlier in Indian mathematics and were known in various cultures, but Fibonacci popularized their study in the Western world.

Formulas and properties: A closed form, known as Binet’s formula, expresses F(n) as (phi^n − psi^n)/√5, where phi is the golden ratio (1+√5)/2 and psi=(1−√5)/2. For large n, F(n) ≈ phi^n/√5. The sum of the first n Fibonacci numbers equals F(n+2) − 1. A common divisibility property is gcd(F_m, F_n) = F_gcd(m,n); in particular, F_k divides F_n iff k divides n. Variants include indexing choices, such as starting with F(1)=1, F(2)=1.

Applications and related concepts: Fib numbers appear in combinatorics, such as counting tilings of a 1×n board with squares and dominoes, where the count is F(n+1). They arise in algorithm design and analysis, including Fibonacci search and dynamic programming. The Fibonacci heap is a data structure named for its amortized growth properties related to the sequence.

See also: Zeckendorf’s representation states that every positive integer can be uniquely represented as the sum of nonconsecutive Fibonacci numbers.