# Fibonacci numbers

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## 5 matching pages

##### 1: 26.11 Integer Partitions: Compositions
The Fibonacci numbers are determined recursively by …
26.11.6 $c\left(\in\!T,n\right)=F_{n-1},$ $n\geq 1$.
26.11.7 $F_{n}=\frac{(1+\sqrt{5})^{n}-(1-\sqrt{5})^{n}}{2^{n}\,\sqrt{5}}.$
Additional information on Fibonacci numbers can be found in Rosen et al. (2000, pp. 140–145).
##### 2: 24.15 Related Sequences of Numbers
###### §24.15(iv) Fibonacci and Lucas Numbers
The Fibonacci numbers are defined by $u_{0}=0$, $u_{1}=1$, and $u_{n+1}=u_{n}+u_{n-1}$, $n\geq 1$. …
24.15.11 $\sum_{k=0}^{\left\lfloor\ifrac{n}{2}\right\rfloor}{n\choose 2k}\left(\frac{5}{% 9}\right)^{k}B_{2k}u_{n-2k}=\frac{n}{6}v_{n-1}+\frac{n}{3^{n}}v_{2n-2},$
For further information on the Fibonacci numbers see §26.11.
##### 3: Bibliography K
• R. P. Kelisky (1957) On formulas involving both the Bernoulli and Fibonacci numbers. Scripta Math. 23, pp. 27–35.
• ##### 4: 18.5 Explicit Representations
In (18.5.4_5) see §26.11 for the Fibonacci numbers $F_{n}$. …
##### 5: Bibliography H
• K. Horata (1989) An explicit formula for Bernoulli numbers. Rep. Fac. Sci. Technol. Meijo Univ. 29, pp. 1–6.
• K. Horata (1991) On congruences involving Bernoulli numbers and irregular primes. II. Rep. Fac. Sci. Technol. Meijo Univ. 31, pp. 1–8.
• F. T. Howard (1996a) Explicit formulas for degenerate Bernoulli numbers. Discrete Math. 162 (1-3), pp. 175–185.
• F. T. Howard (1996b) Sums of powers of integers via generating functions. Fibonacci Quart. 34 (3), pp. 244–256.
• I. Huang and S. Huang (1999) Bernoulli numbers and polynomials via residues. J. Number Theory 76 (2), pp. 178–193.