Numeri Fibonacciani (original) (raw)

Quadrata quorum laterum longitudines numeros Fibonaccianos sequuntur.

Numeri Fibonacciani[1] sunt series numerorum. Primus et secundus numerus sunt 0 et 1. Tota series est 0, 1, 1, 2, 3, 5, 8, 13, … Regula est: proximus numerus est summa duorum priorum, vel

a n = a n − 2 + a n − 1 {\displaystyle a_{n}=a_{n-2}+a_{n-1}} {\displaystyle a_{n}=a_{n-2}+a_{n-1}}

Leonardus Pisanus seriem anno 1202 descripsit.

Anno 1843 Iacobus Philippus Maria Binet(d)(en) formulam generalem invenit quae omnes numeros Fibonaccianos gignit:[2][3][4][5]

a n = ( 1 + 5 2 ) n − ( 1 − 5 2 ) n 5 {\displaystyle a_{n}={\frac {\left({\frac {1+{\sqrt {5}}}{2}}\right)^{n}-\left({\frac {1-{\sqrt {5}}}{2}}\right)^{n}}{\sqrt {5}}}} {\displaystyle a_{n}={\frac {\left({\frac {1+{\sqrt {5}}}{2}}\right)^{n}-\left({\frac {1-{\sqrt {5}}}{2}}\right)^{n}}{\sqrt {5}}}}

Tres quantitates constantes in formula Binettiana sunt: 1 + 5 2 ≈ 1.618033989 {\displaystyle {\frac {1+{\sqrt {5}}}{2}}\approx 1.618033989} {\displaystyle {\frac {1+{\sqrt {5}}}{2}}\approx 1.618033989}, 1 − 5 2 ≈ − 0.618033989 {\displaystyle {\frac {1-{\sqrt {5}}}{2}}\approx -0.618033989} {\displaystyle {\frac {1-{\sqrt {5}}}{2}}\approx -0.618033989}, et 5 ≈ 2.236067977 {\displaystyle {\sqrt {5}}\approx 2.236067977} {\displaystyle {\sqrt {5}}\approx 2.236067977}.

  1. ↑ Giovanni Damiani: Perché Fibonacci? (Italiane)
  2. ↑ Weisstein, Ericus Wolfgangus. Binet's Formula. Wolfram MathWorld.
  3. ↑ Livio, Marius (2002). The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. Novi Eboraci: Broadway Books. p. 106.
  4. ↑ Séroul, Raymond (2000). Programming for Mathematicians. Berolini: Springer-Verlag. p. 21.
  5. ↑ Belcastro, Sarah Maria (2018). Discrete Mathematics with Ducks (2a. ed.). Londinii: CRC Press. p. 260. ISBN 978-1-351-68369-2.

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