%I A000204 M2341 N0924
%S A000204 1,3,4,7,11,18,29,47,76,123,199,322,521,843,1364,2207,3571,
%T A000204 5778,9349,15127,24476,39603,64079,103682,167761,271443,439204,
%U A000204 710647,1149851,1860498,3010349,4870847,7881196,12752043
%N A000204 Lucas numbers (beginning with 1): L(n) = L(n-1) + L(n-2) with L(1) =
1, L(2) = 3.
%C A000204 See A000032 for the version beginning 2, 1, 3, 4, 7, ...
%C A000204 L(n) is the number of matchings in a cycle on n vertices: L(4)=7 because
the matchings in a square with edges a,b,c,d (labeled consecutively)
are the empty set,a,b,c,d,ac and bd. - Emeric Deutsch (deutsch(AT)duke.poly.edu),
Jun 18 2001
%C A000204 This comment covers a family of sequences which satisfy a recurrence
of the form a(n) = a(n-1) + a(n-m), with a(n) = 1 for n = 1...m-1,
a(m) = m+1. The generating function is (x+m*x^m)/(1-x-x^m). Also
a(n) = 1 + n*sum(binomial(n-1-(m-1)*i, i-1)/i, i=1..n/m). This gives
the number of ways to cover (without overlapping) a ring lattice
(or necklace) of n sites with molecules that are m sites wide. Special
cases: m=2: A000204, m=3: A001609, m=4: A014097, m=5: A058368, m=6:
A058367, m=7: A058366, m=8: A058365, m=9: A058364.
%C A000204 L(n) is the number of points of period n in the golden mean shift. The
number of orbits of length n in the golden mean shift is given by
the n-th term of the sequence A006206 - Thomas Ward (t.ward(AT)uea.ac.uk),
Mar 13 2001
%C A000204 Row sums of A029635 are 1,1,3,4,7,... - Paul Barry (pbarry(AT)wit.ie),
Jan 30 2005
%C A000204 a(n) counts circular n-bit strings with no repeated 1's. E.g. for a(5):
00000 00001 00010 00100 00101 01000 01001 01010 10000 10010 10100.
Note #{0...} = fib(n+1), #{1...} = fib(n-1), #{000..., 001..., 100...}
= a(n-1), #{010..., 101...} = a(n-2). - Len Smiley (smiley(AT)math.uaa.alaska.edu),
Oct 14 2001
%C A000204 Contribution from Matthew Lehman (matt.comicopia(AT)gmail.com), Nov 17
2008: (Start)
%C A000204 In the Fibonacci sequence, F(n) = F(n-1) + F(n-2),
%C A000204 for every ith number, F(n+i) = A(i)*F(n) + B(i)*F(n-i),
%C A000204 A(i) is given by this sequence,
%C A000204 Also, A(i) = F(2*i-1)/F(i-1).
%C A000204 B(i) = (-1)^(i+1).
%C A000204 For every Fibonacci number, F(n+1) = F(n) + F(n-1).
%C A000204 For every 2nd Fibonacci number, F(n+2) = 3*F(n) - F(n-2).
%C A000204 For every 3rd Fibonacci number, F(n+3) = 4*F(n) + F(n-3).
%C A000204 For every 4th Fibonacci number, F(n+4) = 7*F(n) - F(n-4).
%C A000204 For every 5th Fibonacci number, F(n+5) = 11*F(n) + F(n-5).
%C A000204 (End)
%C A000204 A014217(n+2)-A014217(n). A014217=1,1,2,4,6,11,17,29,. In A014217 L(n),
Lucas, with L(0)=2,L(1)=1, A000032. See submitted A153263. [From
Paul Curtz (bpcrtz(AT)free.fr), Dec 22 2008]
%D A000204 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%D A000204 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973
(includes this sequence).
%D A000204 P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY,
1968, vol. 2, p. 69.
%D A000204 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 46.
%D A000204 E. Di Cera and Y. Kong, Theory of multivalent binding in one and two-dimensional
lattices, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124.
%D A000204 Leonhard Euler, Introductio in analysin infinitorum (1748), sections
216 and 229.
%D A000204 G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence
Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
%D A000204 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers.
3rd ed., Oxford Univ. Press, 1954, p. 148.
%D A000204 V. E. Hoggatt, Jr., Fibonacci and Lucas Numbers. Houghton, Boston, MA,
1969.
%D A000204 Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley
and Sons, 2001.
%D A000204 Yash Puri and Thomas Ward, A dynamical property unique to the Lucas sequence,
Fibonacci Quarterly, Volume 39, Number 5 (November 2001), pp. 398-402.
%D A000204 Mark A. Shattuck and Carl G. Wagner, Periodicity and Parity Theorems
for a Statistic on r-Mino Arrangements, Journal of Integer Sequences,
Vol. 9 (2006), Article 06.3.6.
%D A000204 S. Vajda, Fibonacci and Lucas numbers and the Golden Section, Ellis Horwood
Ltd., Chichester, 1989.
%H A000204 N. J. A. Sloane, <a href="b000204.txt">The first 500 Lucas numbers: Table
of n, L(n) for n = 1..500</a>
%H A000204 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
RecursiveSequences.html">Recursive Sequences</a>
%H A000204 G. Everest, A. J. van der Poorten, Y. Puri and T. Ward, <a href="http:/
/www.cs.uwaterloo.ca/journals/JIS/index.html">Integer Sequences and
Periodic Points</a>, Journal of Integer Sequences, Vol. 5 (2002),
Article 02.2.3
%H A000204 R. Jovanovic, <a href="http://milan.milanovic.org/math/Math.php?akcija=SviLukasovi">
First 70 Lucas numbers</a>
%H A000204 B. Kelly, <a href="http://home.att.net/~blair.kelly/mathematics/fibonacci/
lucas.txt">Factorizations of Lucas numbers</a>
%H A000204 C. Kimberling, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
Matrix Transformations of Integer Sequences</a>, J. Integer Seqs.,
Vol. 6, 2003.
%H A000204 R. D. Knott, <a href="http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/
lucasNbs.html">The Lucas Numbers in Pascal's Triangle</a>.
%H A000204 A. F. Labossiere, <a href="http://members.lycos.co.uk/sobalian/index.html">
Sobalian Coefficients</a>.
%H A000204 A. F. Labossiere, <a href="http://members.lycos.co.uk/stereotomography/
index.html">Miscellaneous</a>.
%H A000204 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al,
1992.
%H A000204 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
1031 Generating Functions and Conjectures</a>, Universit\'{e} du
Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A000204 Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/
index.html">Arithmetic and growth of periodic orbits</a>, J. Integer
Seqs., Vol. 4 (2001), #01.2.1.
%H A000204 N. J. A. Sloane, <a href="a204.html">Illustration of initial terms: the
Lucas tree</a>
%H A000204 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 A000204 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
Lucasn-StepNumber.html">Lucas n-Step Number</a>
%H A000204 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A000204 Expansion of x(1+2x)/(1-x-x^2). - S. Plouffe, dissertation 1992; multiplied
by x by R. J. Mathar, Nov 14 2007]
%F A000204 a(n) = A000045(2n)/A000045(n). - Benoit Cloitre (benoit7848c(AT)orange.fr),
Jan 05 2003
%F A000204 For n > 1, L(n) = A000045(n+2) - A000045(n-2) (A000045 = Fibonacci numbers)
- Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Jul 10 2004
%F A000204 a(n+1) = 4*A054886(n+3) - A022388(n) - 2*A022120(n+1) (a conjecture;
note that the above sequences have different offsets). Generating
floretion: - 0.25'i - 0.5'k - 0.25i' - 0.5j' - 0.5k' - 0.75'ii' +
0.75'jj' + 0.25'kk' + 0.25'jk' - 0.5'ki' + 0.25'kj' - 0.25e - Creighton
Dement (creighton.k.dement(AT)uni-oldenburg.de), Nov 27 2004
%F A000204 L(n) = (1/(n-1)!) * [ n^(n-1) - { -C(n-2, 0) + 2*C(n-2, 1) + 3*C(n-2,
2) }*n^(n-2) + { 2*C(n-3, 0) + 15*C(n-3, 1) + 51*C(n-3, 2) + 65*C(n-3,
3) + 27*C(n-3, 4) }*n^(n-3) - { -6*C(n-4, 0) + 148*C(n-4, 1) + 945*C(n-4,
2) + 2292*C(n-4, 3) + 2776*C(n-4, 4) + 1680*C(n-4, 5) + 405*C(n-4,
6) }*n^(n-4) + ..... ]. - Andre F. Labossiere (boronali(AT)laposte.net),
Nov 30 2004
%F A000204 a(n)=sum{k=0..floor((n+1)/2), (n+1)*binomial(n-k+1, k)/(n-k+1)} - Paul
Barry (pbarry(AT)wit.ie), Jan 30 2005
%F A000204 L(n+1) = A000045(n+4) - 2*A000045(n+1) - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de),
Oct 07 2005
%F A000204 L(n) = (1/sqrt(5))*(2.5+0.5*sqrt(5))*(0.5+0.5*sqrt(5))^n + (1/sqrt(5))*(-2.5+0.5*sqrt(5))*(0.5-0.5*sqrt(5))^n\
. - Antonio A. Olivares (olivares14031(AT)yahoo.com), Feb 28 2006
%F A000204 L(n) = A000045(n+1) + A000045(n-1). - John Blythe Dobson (j.dobson(AT)uwinnipeg.ca),
Sep 29 2007
%F A000204 a(n)=2*fibonacci(n-1)+fibonacci(n), n>=1 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Oct 05 2007
%F A000204 L(n) = term (1,1) in the 1x2 matrix [2,-1].[1,1; 1,0]^n. - Alois P. Heinz
(heinz(AT)hs-heilbronn.de), Jul 25 2008
%F A000204 a(n)=G^n+(1-G)^n = G^n+(-G)^(-n) where G is Golden ratio = (1+Sqrt[5])/
2 = 1.618033989 [From Artur Jasinski (grafix(AT)csl.pl), Oct 05 2008]
%F A000204 a(n)=((1+sqrt5)^n-(1-sqrt5)^n)/(2^n*sqrt5)+((1+sqrt5)^(n-1)-(1-sqrt5)^(n-1))/
(2^(n-2)*sqrt5). Offset 1. a(3)=4. [From Al Hakanson (hawkuu(AT)gmail.com),
Jan 12 2009, Jan 14 2009]
%p A000204 A000204 := proc(n) option remember; if n <=2 then 2*n-1; else A000204(n-1)+A000204(n-2);
fi; end;
%p A000204 with(combinat): A000204 := n->fibonacci(n+1)+fibonacci(n-1); # an alternative
program
%p A000204 L[1]:=1: L[2]:=3: for n from 3 to 34 do L[n]:=L[n-1]+L[n-2] od:seq(L[n],
n=1..34);
%p A000204 a:=n->2*fibonacci(n-1)+fibonacci(n): seq(a(n), n=1..34); - Zerinvary
Lajos (zerinvarylajos(AT)yahoo.com), Oct 05 2007
%p A000204 A000204:=-z*(1+2*z)/(-1+z+z**2); [S. Plouffe in his 1992 dissertation.]
%p A000204 L := n -> (Matrix([[2,-1]]).Matrix ([[1,1], [1,0]])^n)[1,1]; seq (L(n),
n=1..50); - Alois P. Heinz (heinz(AT)hs-heilbronn.de), Jul 25 2008
%t A000204 c = (1 + Sqrt[5])/2; Table[Expand[c^n + (1-c)^n], {n, 1, 30}] [From Artur
Jasinski (grafix(AT)csl.pl), Oct 05 2008]
%t A000204 Table[LucasL[n, 1], {n, 1, 36}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Jul 09 2009]
%o A000204 (PARI) A000204(n)=fibonacci(n+1)+fibonacci(n-1) [From Michael Porter
(michael_b_porter(AT)yahoo.com), Nov 05 2009]
%Y A000204 Cf. A000032, A000045, A061084, A027960.
%Y A000204 Cf. also A001609, A014097, A000079, A003269, A003520, A005708, A005709,
A005710, A006206.
%Y A000204 Cf. A101033, A101032, A100492, A099731, A094216, A094638, A000108.
%Y A000204 Sequence in context: A147869 A100581 A093090 this_sequence A075193 A042433
A024319
%Y A000204 Adjacent sequences: A000201 A000202 A000203 this_sequence A000205 A000206
A000207
%K A000204 core,easy,nonn,nice,new
%O A000204 1,2
%A A000204 N. J. A. Sloane (njas(AT)research.att.com).
%E A000204 Additional comments from Yong Kong (ykong(AT)curagen.com), Dec 16 2000
%E A000204 More terms from Andre F. Labossiere (boronali(AT)laposte.net), Nov 30
2004
%E A000204 Plouffe Maple line edited by N. J. A. Sloane (njas(AT)research.att.com),
May 13 2008
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