%I A000032 M0155
%S A000032 2,1,3,4,7,11,18,29,47,76,123,199,322,521,843,1364,2207,3571,5778,
%T A000032 9349,15127,24476,39603,64079,103682,167761,271443,439204,710647,
%U A000032 1149851,1860498,3010349,4870847,7881196,12752043,20633239,33385282
%N A000032 Lucas numbers (beginning at 2): L(n) = L(n-1) + L(n-2). (Cf. A000204.)
%C A000032 This is also the Horadam sequence (2,1,1,1). - Ross La Haye (rlahaye(AT)new.rr.com),
Aug 18 2003
%C A000032 For distinct primes p,q, L(p) is congruent to 1 mod p, L(2p) is congruent
to 3 mod p and L(pq) is congruent 1+q(L(q)-1) mod p. Also, L(m) divides
F(2km) and L((2k+1)m), k,m >=0.
%C A000032 a(n)=sum(P(3;n-1-k,k),k=0..ceiling((n-1)/2)), n>=1, with a(0)=2. These
are the sums over the SW-NE diagonals in P(3;n,k), the (3,1) Pascal
triangle A093560. Observation by Paul Barry (pbarry(AT)wit.ie), Apr
29 2004. Proof via recursion relations and comparison of inputs.
Also SW-NE diagonal sums of the (1,2) Pascal triangle A029635 (with
T(0,0) replaced by 2).
%C A000032 Suppose psi=ln(phi). We get the representation L(n)=2*cosh(n*psi) if
n is even; L(n)=2*sinh(n*psi) if n is odd. There is a similar representation
for Fibonacci numbers (A000045). Many Lucas formulas now easily follow
from appropriate sinh- and cosh-formulas. For example: the identity
cosh^2(x)-sinh^2(x)=1 implies L(n)^2-5F(n)^2=4*(-1)^n (setting x=n*psi).
- Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Apr 18 2007
%C A000032 Comments from John Blythe Dobson (j.dobson(AT)uwinnipeg.ca), Oct 02 2007,
Oct 11 2007: (Start) The parity of L(n) follows easily from its definition,
which shows that L(n) is even when n is a multiple of 3 and odd otherwise.
%C A000032 The first six multiplication formulae are:
%C A000032 L(2n) = (L(n))^2 - 2*(-1)^n
%C A000032 L(3n) = (L(n))^3 - 3*((-1)^n)*L(n)
%C A000032 L(4n) = (L(n))^4 - 4*((-1)^n)*(L(n))^2 + 2
%C A000032 L(5n) = (L(n))^5 - 5*((-1)^n)*(L(n))^3 + 5*L(n)
%C A000032 L(6n) = (L(n))^6 - 6*((-1)^n)*(L(n))^4 + 9*(L(n))^2 - 2*(-1)^n
%C A000032 Generally, L(n) | L(mn) iff m is odd. (End)
%C A000032 In the expansion of L(mn), where m represents the multiplier and n the
index of a known value of L(n), the absolute values of the coefficients
are the terms in the m-th row of the triangle A034807. When m=1 and
n=1, L(n)=1 and all the terms are positive and so the row sums of
A034807 are simply the Lucas numbers. (End)
%C A000032 The comments submitted by Miklos Kristof on Mar 19 2007 for the Fibonacci
numbers (A00045) contain four important identities which have close
analogues in the Lucas numbers: For a>=b and odd b, L(a+b) + L(a-b)
= 5*F(a)*F(b). For a>=b and even b, L(a+b) + L(a-b) = L(a)*L(b).
For a>=b and odd b, L(a+b) - L(a-b) = L(a)*L(b). For a>=b and even
b, L(a+b) - L(a-b) = 5*F(a)*F(b). - John Blythe Dobson (j.dobson(AT)uwinnipeg.ca),
Nov 15 2007. A particularly interesting instance of the difference
identity for even b is L(a+30) - L(a-30) = 5*F(a)*832040, since 5*832040
is divisible by 100, proving that the last two digits of Lucas nu
mbers repeat in a cycle of length 60.
%C A000032 Further comments from John Blythe Dobson (j.dobson(AT)uwinnipeg.ca),
Nov 15 2007: (Start) The Lucas numbers satisfy remarkable difference
equations, in some cases best expressed using Fibonacci numbers,
of which representative examples are the following:
%C A000032 L(n) - L(n-3) = 2*L(n-2)
%C A000032 L(n) - L(n-4) = 5*F(n-2)
%C A000032 L(n) - L(n-6) = 4*L(n-3)
%C A000032 L(n) - L(n-12) = 40*F(n-6)
%C A000032 L(n) - L(n-60) = 4160200*F(n-30).
%C A000032 These formulae establish, respectively, that the Lucas numbers form a
cyclic residue system of length 3 (mod 2), of length 4 (mod 5), of
length 6 (mod 4), of length 12 (mod 40) and of length 60 (mod 4160200).
The divisibility of the last modulus by 100 accounts for the fact
that the last two digits of the Lucas numbers begin to repeat at
L(60).
%C A000032 The divisibility properties of the Lucas numbers are very complex and
still not fully understood, but several important criteria are established
in Zhi-Hong Sun's 2003 survey of congruences for Fibonacci numbers.
(End)
%C A000032 Sum_{n>0} a(n)/(n*2^n) = 2*log(2) [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com),
Oct 11 2009]
%D A000032 P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY,
1968, vol. 2, p. 69.
%D A000032 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of
combinatorial proof, M.A.A. 2003, id. 32,50.
%D A000032 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 46.
%D A000032 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers.
3rd ed., Oxford Univ. Press, 1954, p. 148.
%D A000032 V. E. Hoggatt, Jr., Fibonacci and Lucas Numbers. Houghton, Boston, MA,
1969.
%D A000032 Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley
and Sons, 2001.
%D A000032 A. McLeod and W. O. J. Moser, Counting cyclic binary strings, Math. Mag.,
80 (No. 1, 2007), 29-37.
%D A000032 Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas
n-step Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article
05.4.4.
%D A000032 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%D A000032 Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the
Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article
06.1.1.
%D A000032 S. Vajda, Fibonacci and Lucas numbers and the Golden Section, Ellis Horwood
Ltd., Chichester, 1989.
%H A000032 N. J. A. Sloane, <a href="b000032.txt">The first 500 Lucas numbers: Table
of n, L(n) for n = 0..500</a>
%H A000032 G. Everest, Y. Puri and T. Ward, <a href="http://arXiv.org/abs/math.NT/
0204173">Integer sequences counting periodic points</a>
%H A000032 R. Javonovic, <a href="http://milan.milanovic.org/math/english/function1/
function1.html">Lucas Function Calculator</a>
%H A000032 B. Kelly, <a href="http://home.att.net/~blair.kelly/mathematics/fibonacci/
lucas.txt">Factorizations of Lucas numbers</a>
%H A000032 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
RecursiveSequences.html">Recursive Sequences</a>
%H A000032 R. Knott, <a href="http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/
lucasNbs.html">The Lucas numbers</a>
%H A000032 R. Knott, <a href="http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/
lucas200.html">The First 200 Lucas numbers and their factors</a>
%H A000032 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/
matha1/matha113.htm">Factorizations of many number sequences</a>
%H A000032 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/
matha1/matha114.htm">Factorizations of many number sequences</a>
%H A000032 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/
matha1/matha115.htm">Factorizations of many number sequences</a>
%H A000032 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/
matha1/matha116.htm">Factorizations of many number sequences</a>
%H A000032 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/
matha1/matha117.htm">Factorizations of many number sequences</a>
%H A000032 B. Rittaud, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL10/Rittaud2/
rittaud11.pdf">On the Average Growth of Random Fibonacci Sequences</
a>, Journal of Integer Sequences, 10 (2007), Article 07.2.4.
%H A000032 Zhi-Hong Sun, <a href="http://202.195.112.2/xsjl/szh/ConFn.pdf">Congruences
for Fibonacci Numbers</a> [PDF] (Lecture notes, 2003)
%H A000032 Dan Sewell Ward, <a href="http://www.halexandria.org/dward094.htm">Modified
Fibonacci Sequence</a>.
%H A000032 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
LucasNumber.html">Link to a section of The World of Mathematics.</
a>
%H A000032 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to
Chebyshev polynomials.</a>
%H A000032 <a href="Sindx_Rea.html#recur1">Index entries for recurrences a(n) =
k*a(n - 1) +/- a(n - 2)</a>
%H A000032 <a href="Sindx_Tu.html#2wis">Index entries for two-way infinite sequences</
a>
%H A000032 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to
linear recurrences with constant coefficients</a>
%F A000032 Conjecture: Let f(n) = Phi^n + Phi^(-n), then L(2n) = f(2n) and L(2n+1)
= f(2n+1) - 2*Sum(k=0..infinity, C(k+1)/f(2n+1)^(2k+1)) where C(n)
are Catalan numbers (A000108). - Gerald McGarvey (gerald.mcgarvey(AT)comcast.net),
Dec 21 2007
%F A000032 G.f.: (2-x)/(1-x-x^2). L(n)=((1+sqrt(5))/2)^n + ((1-sqrt(5))/2)^n.
%F A000032 L(n) = L(n-1) + L(n-2) = (-1)^n L(-n).
%F A000032 E.g.f.: 2*exp(x/2)*cosh(sqrt(5)*x/2). - Len Smiley (smiley(AT)math.uaa.alaska.edu),
Nov 30 2001
%F A000032 L(n) = Fibonacci(2n)/Fibonacci(n) [ Jeff Burch (gburch(AT)erols.com)
]
%F A000032 L(n) = Fib(n) + 2*Fib(n-1) = Fib(n + 1) + Fib(n-1) - Henry Bottomley
(se16(AT)btinternet.com), Apr 12 2000
%F A000032 a(n)=sqrt(F(n-1)^2+4*F(n)*F(n-2)) - Benoit Cloitre (benoit7848c(AT)orange.fr),
Jan 06 2003
%F A000032 a(n)=2^(1-n)sum{k=0..floor(n/2), C(n, 2k)5^k}. a(n)=2T(n, i/2)(-i)^n
with T(n, x) Chebyshev's polynomials of the first kind (see A053120)
and i^2=-1. - Paul Barry (pbarry(AT)wit.ie), Nov 15 2003
%F A000032 L(n)=2*Fib(n+1)-Fib(n) - Paul Barry (pbarry(AT)wit.ie), Mar 22 2004
%F A000032 a(n)=floor((phi)^n+(-phi)^(-n)) - Paul Barry (pbarry(AT)wit.ie), Mar
12 2005
%F A000032 Comments from Miklos Kristof (kristmikl(AT)freemail.hu), Mar 19 2007:
(Start)
%F A000032 Let F(n)=A000045=Fibonacci numbers, L(n)=a(n)=Lucas numbers:
%F A000032 L(n+m)+(-1)^m*L(n-m)=L(n)*L(m)
%F A000032 L(n+m)-(-1)^m*L(n-m)=8*F(n)*F(m)
%F A000032 L(n+m+k)+(-1)^k*L(n+m-k)+(-1)^m*(L(n-m+k)+(-1)^k*L(n-m-k))=L(n)*L(m)*L(k)
%F A000032 L(n+m+k)-(-1)^k*L(n+m-k)+(-1)^m*(L(n-m+k)-(-1)^k*L(n-m-k))=5*F(n)*L(m)*F(k)
%F A000032 L(n+m+k)+(-1)^k*L(n+m-k)-(-1)^m*(L(n-m+k)+(-1)^k*L(n-m-k))=5*F(n)*F(m)*L(k)
%F A000032 L(n+m+k)-(-1)^k*L(n+m-k)-(-1)^m*(L(n-m+k)-(-1)^k*L(n-m-k))=5*L(n)*F(m)*F(k)
(End)
%F A000032 Inverse: floor(log_phi(a(n))+0.5)=n, for n>1. Also for n>=0, floor(1/
2*log_phi(a(n)*a(n+1)))=n. Extension valid for all integers n: floor(1/
2*sign(a(n)*a(n+1))*log_phi|a(n)*a(n+1)|)=n {where sign(x) = sign
of x}. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), May 02
2007
%F A000032 Starting (1, 3, 4, 7, 11,...) = row sums of triangle A131774. - Gary
W. Adamson (qntmpkt(AT)yahoo.com), Jul 14 2007
%F A000032 a(n)=2*fibonacci(n-1)+fibonacci(n), n>=0 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Oct 05 2007
%F A000032 a(n) = trace of the 2 X 2 matrix [0,1; 1,1]^n - Gary W. Adamson (qntmpkt(AT)yahoo.com),
Mar 02 2008
%F A000032 Comments from Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jan
02 2009 (Start): For odd n: a(n)=floor(1/(fract(phi^n))); for even
n>0: a(n)=ceiling(1/(1-fract(phi^n))). This follows from the basic
property of the golden ratio phi, which suffices phi-phi^(-1)=1 (see
general formula described in A001622).
%F A000032 a(n)=nint(1/(min(fract(phi^n), 1-fract(phi^n))), for n>1, where fract(x)=x-floor(x).
(End)
%p A000032 with(combinat): A000032 := n->fibonacci(n+1)+fibonacci(n-1);
%p A000032 a:=n->2*fibonacci(n-1)+fibonacci(n): seq(a(n), n=0..36); - Zerinvary
Lajos (zerinvarylajos(AT)yahoo.com), Oct 05 2007
%t A000032 a[0] := 2; a[n] := Nest[{Last[ # ], First[ # ] + Last[ # ]} &, {2, 1},
n] // Last
%t A000032 Array[2 Fibonacci[ #+1] - Fibonacci[ # ] &, 50, 0] - Joseph Biberstine
(jrbibers(AT)indiana.edu), Dec 26 2006
%t A000032 Table[LucasL[n, 1], {n, 0, 36}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Jul 09 2009]
%t A000032 a=1;lst={2,a};s=5;Do[a=s-(a+1);AppendTo[lst,a];s+=a,{n,5!}];lst [From
Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 27 2009]
%o A000032 (MAGMA) [ Lucas(n) : n in [0..120]];
%o A000032 (PARI) a(n)=if(n<0,(-1)^n*a(-n),if(n<2,2-n,a(n-1)+a(n-2)))
%o A000032 (PARI) a(n)=if(n<0,(-1)^n*a(-n),polsym(x^2-x-1,n)[n+1])
%o A000032 (PARI) a(n)=real((2+quadgen(5))*quadgen(5)^n)
%o A000032 sage: [lucas_number2(n,1,-1) for n in range(37)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Jun 25 2008
%Y A000032 Cf. A000204. A000045(n)=(2L(n+1)-L(n))/5.
%Y A000032 First row of array A103324.
%Y A000032 a(n) = A101220(2, 0, n), for n > 0.
%Y A000032 a(k) = A090888(1, k) = A109754(2, k) = A118654(2, k-1), for k > 0.
%Y A000032 Cf. A131774.
%Y A000032 Cf. A001622, A006497, A080039.
%Y A000032 Sequence in context: A058658 A070827 A160191 this_sequence A061084 A055391
A134876
%Y A000032 Adjacent sequences: A000029 A000030 A000031 this_sequence A000033 A000034
A000035
%K A000032 nonn,nice,easy,core
%O A000032 0,1
%A A000032 N. J. A. Sloane (njas(AT)research.att.com).
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