Fibonacci numbers

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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

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Symbols:: $\in$ : element of, $c\left(\NVar{\mathrm{condition}},\NVar{n}\right)$ : restricted number of compositions of $n$ , $n$ : nonnegative integer, $T$ : set and $F_{n}$ : Fibonacci numbers
Permalink:: http://dlmf.nist.gov/26.11.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §26.11 and Ch.26

… ►

26.11.7

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

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Symbols:: $n$ : nonnegative integer and $F_{n}$ : Fibonacci numbers
Referenced by:: (18.5.4_5)
Permalink:: http://dlmf.nist.gov/26.11.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §26.11 and Ch.26

►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},

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $k$ : integer, $n$ : integer, $u_{n}$ : Fibonacci numbers and $v_{n}$ : Lucas numbers
Permalink:: http://dlmf.nist.gov/24.15.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §24.15(iv), §24.15 and Ch.24

… ►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.

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External Links:: MathReview (L. Carlitz), ZentralBlatt
Referenced by:: §24.15(iv).
Encodings:: BibTeX

…

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.

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External Links:: ZentralBlatt
Referenced by:: §24.6.
Encodings:: BibTeX

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K. Horata (1991) On congruences involving Bernoulli numbers and irregular primes. II. Rep. Fac. Sci. Technol. Meijo Univ. 31, pp. 1–8.

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External Links:: ZentralBlatt
Referenced by:: §24.15(iii).
Encodings:: BibTeX

… ►

F. T. Howard (1996a) Explicit formulas for degenerate Bernoulli numbers. Discrete Math. 162 (1-3), pp. 175–185.

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External Links:: ISSN 0012-365X, Document, MathReview (L. Skula), ZentralBlatt
Referenced by:: §24.16(i).
Encodings:: BibTeX

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F. T. Howard (1996b) Sums of powers of integers via generating functions. Fibonacci Quart. 34 (3), pp. 244–256.

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External Links:: ISSN 0015-0517, MathReview (Noriaki Kimura), ZentralBlatt
Referenced by:: §24.4(iii).
Encodings:: BibTeX

… ►

I. Huang and S. Huang (1999) Bernoulli numbers and polynomials via residues. J. Number Theory 76 (2), pp. 178–193.

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External Links:: ISSN 0022-314X, Document, MathReview (L. Skula), ZentralBlatt
Referenced by:: §24.14(ii).
Encodings:: BibTeX

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