Fibonacci numbers

Fibonacci numbers are defined by the recurrence $F_1=1$

and $F_n=F_{n-1}+F_{n-2}$ for

2$">.

Their closed form is

$F_n = \frac{(1+\sqrt{5})^n-(1-\sqrt{5})^n}{2^n\sqrt{5}}\,.$

An interesting sum for any integer $\alpha\ge2$ is

$\sum_{k=1}^{\infty} \frac{F_k}{\alpha^k} = \frac{\alpha}{\alpha^2-\alpha-1}\,.$

The first Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946 more terms

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here

A graph displaying how many Fibonacci numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.

Useful links Mathworld, Fibonacci Number
Wikipedia, Fibonacci number
OEIS, Sequence A000045

Fibonacci numbers can also be... (you may click on names or numbers and on + to get more values)

a-pointer 13 aban 13 21 34 55 89 144 233 377 610 987 abundant 144 2584 46368 832040 14930352 alternating 21 34 89 610 987 4181 6765 amenable 13 21 89 144 233 377 1597 2584 4181 + 267914296 433494437 701408733 apocalyptic 610 4181 6765 10946 17711 28657 arithmetic 13 21 55 89 233 377 987 1597 4181 + 2178309 5702887 9227465 binomial 21 55 brilliant 21 377 121393 5702887 c.nonagonal 55 c.square 13 c.triangular 2584 Chen 13 89 233 514229 congruent 13 21 34 55 987 1597 2584 4181 6765 + 2178309 5702887 9227465 constructible 34 Curzon 21 89 233 4181 10946 196418 cyclic 13 89 233 377 1597 4181 17711 28657 121393 514229 1346269 5702887 D-number 21 d-powerful 89 377 de Polignac 1597 121393 514229 deficient 13 21 34 55 89 233 377 610 987 + 3524578 5702887 9227465 dig.balanced 21 14930352 Duffinian 21 55 144 377 4181 17711 75025 121393 1346269 5702887 9227465 eban 34 economical 13 21 89 233 1597 17711 28657 121393 514229 1346269 emirp 13 1597 emirpimes 4181 17711 1346269 equidigital 13 21 89 233 1597 17711 28657 121393 514229 1346269 esthetic 21 34 89 987 6765 evil 34 89 144 377 610 987 2584 6765 10946 + 267914296 433494437 701408733 Friedman 46368 gapful 46368 14930352 102334155 267914296 1134903170 4807526976 good prime 1597 happy 13 2584 4181 75025 2178309 3524578 Harshad 21 144 2584 14930352 heptagonal 34 55 hoax 1346269 Hogben 13 21 hungry 144 hyperperfect 21 iban 21 144 377 17711 iccanobiF 13 idoneal 13 21 inconsummate 377 317811 514229 interprime 21 34 144 610 987 24157817 Jacobsthal 21 Jordan-Polya 144 junction 610 75025 Kaprekar 55 katadrome 21 610 987 lucky 13 21 1597 6765 75025 magic 34 magnanimous 21 34 89 metadrome 13 34 89 modest 13 89 233 Moran 21 Motzkin 21 nialpdrome 21 55 610 987 nude 55 144 46368 oban 13 55 89 377 610 987 octagonal 21 odious 13 21 55 233 1597 4181 28657 121393 196418 + 3524578 9227465 165580141 palindromic 55 pancake 1597 panconsummate 21 34 89 144 pandigital 21 pernicious 13 21 34 55 144 233 1597 4181 28657 + 1346269 3524578 9227465 Pierpont 13 plaindrome 13 34 55 89 144 233 377 power 144 powerful 144 practical 144 46368 832040 prim.abundant 2584 prime 13 89 233 1597 28657 514229 433494437 2971215073 primeval 13 Proth 13 pseudoperfect 144 2584 46368 832040 repdigit 55 repunit 13 21 self 233 17711 121393 1346269 semiprime 21 34 55 377 4181 17711 121393 1346269 5702887 Smith 1346269 Sophie Germain 89 233 sphenic 610 987 10946 3524578 9227465 24157817 39088169 63245986 square 144 star 13 straight-line 987 strong prime 1597 28657 super-d 4181 1346269 9227465 tau 46368 triangular 21 55 tribonacci 13 truncatable prime 13 233 twin 13 uban 13 21 55 89 Ulam 13 2584 514229 unprimeable 2584 46368 196418 832040 3524578 9227465 untouchable 10946 46368 upside-down 55 wasteful 34 55 144 377 610 987 2584 4181 6765 + 3524578 5702887 9227465 weak prime 13 89 233 514229 Zuckerman 144 Zumkeller 2584 46368 zygodrome 55