Lucas numbers

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1: 24.15 Related Sequences of Numbers

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§24.15(iv) Fibonacci and Lucas Numbers

… ►The Lucas numbers are defined by

v_{0}=2

v_{1}=1

, and

v_{n+1}=v_{n}+v_{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

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24.15.12

\sum_{k=0}^{\left\lfloor\ifrac{n}{2}\right\rfloor}{n\choose 2k}\left(\frac{5}{% 4}\right)^{k}E_{2k}v_{n-2k}=\frac{1}{2^{n-1}}.

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Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $k$ : integer, $n$ : integer and $v_{n}$ : Lucas numbers
Permalink:: http://dlmf.nist.gov/24.15.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §24.15(iv), §24.15 and Ch.24

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2: 27.22 Software

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•

Mathematica. PrimeQ combines strong pseudoprime tests for the bases 2 and 3 and a Lucas pseudoprime test. No known composite numbers pass these three tests, and Bleichenbacher (1996) has shown that this combination of tests proves primality for integers below $10^{16}$ . Provable PrimeQ uses the Atkin–Goldwasser–Kilian–Morain Elliptic Curve Method to prove primality. FactorInteger tries Brent–Pollard rho, Pollard $p-1$ , and then cfrac after trial division. See §27.19. ecm is available also, and the Multiple Polynomial Quadratic sieve is expected in a future release.

For additional Mathematica routines for factorization and primality testing, including several different pseudoprime tests, see Bressoud and Wagon (2000).

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3: 24.1 Special Notation

… ►The notations

E_{n}

E_{n}\left(x\right)

, as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …

4: Bibliography L

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D. J. Leeming (1977) An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.

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External Links:: MathReview (D. H. Lehmer), ZentralBlatt
Referenced by:: §24.11.
Encodings:: BibTeX

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O. Lehto and K. I. Virtanen (1973) Quasiconformal Mappings in the Plane. 2nd edition, Springer-Verlag, New York.

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Notes:: Translated from the German by K. W. Lucas, Die Grundlehren der mathematischen Wissenschaften, Band 126
External Links:: MathReview, ZentralBlatt
Referenced by:: §19.9(i).
Encodings:: BibTeX

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É. Lucas (1891) Théorie des nombres. Tome I: Le calcul des nombres entiers, le calcul des nombres rationnels, la divisibilité arithmétique. Gauthier-Villars, Paris (French).

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External Links:: MathReview (W. J. LeVeque), ZentralBlatt
Referenced by:: §24.1, §24.1.
Encodings:: BibTeX

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S. K. Lucas and H. A. Stone (1995) Evaluating infinite integrals involving Bessel functions of arbitrary order. J. Comput. Appl. Math. 64 (3), pp. 217–231.

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External Links:: ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt
Referenced by:: §10.74(vii).
Encodings:: BibTeX

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S. K. Lucas (1995) Evaluating infinite integrals involving products of Bessel functions of arbitrary order. J. Comput. Appl. Math. 64 (3), pp. 269–282.

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External Links:: ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt
Referenced by:: §10.74(vii).
Encodings:: BibTeX

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5: Bibliography R

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H. Rademacher (1973) Topics in Analytic Number Theory. Springer-Verlag, New York.

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Notes:: Edited by E. Grosswald, J. Lehner and M. Newman, Die Grundlehren der mathematischen Wissenschaften, Band 169
External Links:: MathReview (H. Halberstam), ZentralBlatt
Referenced by:: §1.8(iv), §20.7(viii), §20.7(viii), Ch.24.
Encodings:: BibTeX

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S. Ramanujan (1927) Some properties of Bernoulli’s numbers (J. Indian Math. Soc. 3 (1911), 219–234.). In Collected Papers,

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Referenced by:: §24.7(i).
Encodings:: BibTeX

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J. T. Ratnanather, J. H. Kim, S. Zhang, A. M. J. Davis, and S. K. Lucas (2014) Algorithm 935: IIPBF, a MATLAB toolbox for infinite integral of products of two Bessel functions. ACM Trans. Math. Softw. 40 (2), pp. 14:1–14:12.

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External Links:: ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt
Referenced by:: §10.74(vii), §10.74(vii), 7th item, §10.77(ix).
Encodings:: BibTeX

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K. H. Rosen (2004) Elementary Number Theory and its Applications. 5th edition, Addison-Wesley, Reading, MA.

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External Links:: ISBN 0-201-87073-8, MathReview (Norman J. Richert)
Referenced by:: §27.17.
Encodings:: BibTeX

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