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corner top left Fibonacci's and related Number Calculators

Find any Fibonacci's Number

"How many pairs of
rabbits can be bred from
one pair in a year?"
L. Fibonacci
fibonacci spiral Fibonacci calculator
Recursive numbers
Fibonacci's life
Books and links

The Fibonacci Series or the chrysodromos (chrysodromos, lit. the "golden course") is a sequence of numbers first created by the Italian mathematician Leonardo di Pisa, or Pisano, known also under the name Fibonacci in 1202. It is a deceptively simple series, but its ramifications and applications are nearly limitless.

fibonacci series

Fibonacci's Calculator

In mathematics, the Fibonacci numbers form a sequence defined recursively by:

Fn = the n-th Fibonacci number

Fo = 0

F1 = F2 = 1

Fn = Fn-1 + Fn-2

F(-n) = (-1)n-1 · Fn

In words: you start with 0 and 1, and then produce the next Fibonacci number (Fn) by adding the two previous Fibonacci numbers:


n = rank, Fn = corresponding Fibonacci number
n ... -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 ...
Fn ... -8 5 -3 2 -1 1 0 1 1 2 3 5 8 13 21 34 ...
Generate more F numbers

Some interesting F sum identities:

F0 + F1 + F2 + F3 + ... + Fn = Fn+2 - 1

1F1 + 2F2 + 3F3 + ... + nFn = nFn+2 - Fn+3 + 2

Other interesting F properties:

Lucas' Theorem:
Fm gcd Fn = F(m gcd n)

[ gcd = greatest common divisor ]

Cassini's Formula:
Fn+1 · Fn-1 - (Fn)2 = (-1)n

A variant:
Fn-2 · Fn+1 - Fn-1 · Fn = (-1)n-1

Simson's Relation:
Fn+1 · Fn-1 + (-1)n-1 = (Fn)2

Shifting Property:
Fm+n = Fm · Fn+1 + Fm-1 · Fn
The number Phi (phi = radix5/2 + 0.5 ~ 1.618...), or Golden Ratio, is intimately related to Fibonacci numbers. The closed form of Fn is:
Fn = (phin - (1 - phi)n) / radix5
Fn also appears in the expansion of phin. For all n, it is:
phin = Fn-1 + Fn · phi

Try it: For n = , phin = + · phi

Recursive Numbers Calculator TOP

This calculator generates recursive series (like Fibonacci and related numbers), that means a series of numbers where each term is the sum of its 2 predecessors. You just have to determine the first 2 numbers and how many terms you want to have indicated.

1st number
2nd number
Number of terms
Click to show series

Recursive Numbers Family
Fibonacci, Lucas, golden number, silver number Padovan, Perrin, plastic number

Applications TOP

  The Fibonacci numbers are important in the run-time analysis of Euclid's algorithm to determine the greatest common divisor of two integers.
  The Fibonacci numbers occur in a formula about the diagonals of Pascal's triangle (binomial coefficient).

pascal triangle

  In music Fibonacci numbers are sometimes used to determine tunings, and, as in visual art, to determine the length or size of content or formal elements.
  An interesting use of the Fibonacci sequence is for converting miles to kilometers. For instance, if you want to know how many kilometers 8 miles is, take the Fibonacci number (8) and look at the next one (13). 8 miles is about 13 kilometers. This works because it so happens that the conversion factor between miles and kilometers (1.609) is roughly equal to phi (1.618).
  Even more amazing is a surprising relationship to magic squares. Magic squares are arrangements of numbers in a square pattern in which all the rows, columns, and diagonals add up to the same number. The simplest is the 3x3 pattern shown below:

2 7 6
9 5 1
4 3 8

  If one substitutes for these numbers the corresponding Fibonacci number, a new "magic square" is produced in which the sum of the products of the three rows is equal to the sum of the products of the three columns!

Fibonacci (Leonardus Pisanus de filiis Bonaccii) TOP

Fibonacci portrait  The "greatest European mathematician of the middle ages", his full name was Leonardo of Pisa, or Leonardo Pisano in Italian since he was born in Pisa (Italy), the city with the famous Leaning Tower, about 1175 AD. Strangely, Fibonacci is best remembered by the sequence which bears his name but which, ironically, he treated only lightly!
  Little is known about Leonardo's life beyond the few facts given in his writings. During Leonardo's boyhood his father, Guglielmo, an official of the Republic of Pisa, was appointed consul over the community of Pisan merchants in the North African port of Bugia (now Bejaïa, Algery). He intended for Leonardo to become a merchant and so arranged for his instruction in calculational techniques. Leonardo was sent to study calculation with an Arab master. He later went to Egypt, Syria, Greece, Sicily, and Provence, where he studied different numerical systems and methods of calculation.
  Around 1200, Fibonacci returned to Pisa where, for at least the next twenty-five years, he worked on his own mathematical compositions. The five works from this period which have come down to us are:
Liber Abaci, ~1202, 1228. (The Book of Calculating). An encyclopaedia of thirteenth-century mathematics, both theoretical and practical. One of the problems in this book involves the famous sequence 1, 2, 3, 5, 8, 13, ... with which his name is irrevocably linked (Quot paria coniculorum in uno anno ex uno paro germinentur): "A certain man put a pair of rabbits in a place surrounded by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?". Actually, it was much later (~ 1870) that Edouard Lucas named this famous series of numbers after Fibonacci.
arrow You can acquire the English translation of 'Liber Abaci' here.
De Practica Geometriae, ~1220. (Practice of Geometry). A book on geometry based on Euclid's "Elements" and "On Divisions".
Flos, ~1225. (Flower). In this short work (the title of which might suggest that algebra is the "flower of Mathematics"), Fibonacci describes inter alia two of the 'Diophantine problems' he worked on at the court of the Emperor Frederick II.
Epistola ad Magistrum Theodorum, ~1225. A letter to Master Theodorus, the imperial philosopher to the court of the Hohenstaufen emperor Frederick II.
Liber Quadratorum, ~1225. (The Book of Squares). His largest work, a number-theoretical book concerned with the simultaneous solution of quadratic equations in two or more variables.
  These however are not the only books written by Fibonacci. Other works known to have existed include the Di minor guisa, a book for commercial arithmetic.

  So great was Fibonacci's reputation as a mathematician as a result of these works that Frederick II summoned him for an audience when he was in Pisa around 1225.
  After 1228, virtually nothing is known of Fibonacci's life, except that by decree the Republic of Pisa awarded the "'serious and learned Master Leonardo Bigollo' (discretus et sapiens) a yearly salarium of 'libre XX denariorem' in addition to the usual allowances". This stipend rewarded Fibonacci for his pro bono advising to the Republic on matters involving accounting and related mathematical matters.

His various names
  Leonardo Pisano called himself Fibonacci [pronounced fib-on-'ah-chee], short for fillio Bonacci (or Bonaccii), which means "Bonaccio's son" in old Italian (Fi' = "son"; Bonacci(o) + i = "Bonaccio + 's"), since his father's name was Guglielmo Bonaccio. Fi'-Bonacci is like the English name of John-son or the Scottish name Mac-Donald.
  His father's name was most probably a nickname with the ironical meaning of a 'good, stupid fellow', while to Leonardo himself another nickname, Bigollo or Bigollone ('loafer, wanderer', cf. the Italian word bighellone), has been given.

  'Golden Number' page


fibonacci fun
archimedes journal
Fibonacci Fun
Reproducible activities and projects...

Archimedes Journal nr. 1
Learn how to make paradoxical mechanical puzzles using the Fibonacci sequence!

arrow Zoo of numbers: interesting facts about numbers
arrow A very interesting site on Fibonacci sequence
arrow The story of the 'Liber Abaci'
arrow Official website of the Fibonacci Association


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© 2003 G. Sarcone,

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