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%I M2341 N0924
%S 1,3,4,7,11,18,29,47,76,123,199,322,521,843,1364,2207,3571,5778,9349,
%T 15127,24476,39603,64079,103682,167761,271443,439204,710647,1149851,
%U 1860498,3010349,4870847,7881196,12752043,20633239,33385282,54018521,87403803,141422324
%N Lucas numbers (beginning with 1): L(n) = L(n-1) + L(n-2) with L(1) = 1, L(2) = 3.
%C See A000032 for the version beginning 2, 1, 3, 4, 7, ...
%C Also called Schoute's accessory series (see Jean, 1984). - _N. J. A. Sloane_, Jun 08 2011.
%C 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_, Jun 18 2001
%C 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 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 Row sums of A029635 are 1,1,3,4,7,... - _Paul Barry_, Jan 30 2005
%C 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 Row sums of the triangle in A182579. [_Reinhard Zumkeller_, May 07 2012]
%D Mohammad K. Azarian, Identities Involving Lucas or Fibonacci and Lucas Numbers as Binomial Sums, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 45, 2012, pp. 2221-2227.
%D P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 1968, vol. 2, p. 69.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 46.
%D 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 Leonhard Euler, Introductio in analysin infinitorum (1748), sections 216 and 229.
%D G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 148.
%D V. E. Hoggatt, Jr., Fibonacci and Lucas Numbers. Houghton, Boston, MA, 1969.
%D Sarah H. Holliday and Takao Komatsu, On the Sum of Reciprocal Generalized Fibonacci Numbers, Integers. Volume 11, Issue 4, Pages 441-455, DOI: 10.1515/INTEG.2011.031.
%D R. V. Jean, Mathematical Approach to Pattern and Form in Plant Growth, Wiley, 1984. See p. 5. [From _N. J. A. Sloane_, Jun 08 2011]
%D Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001.
%D Mathilde Noual, Dynamics of Circuits and Intersecting Circuits, in LANGUAGE AND AUTOMATA THEORY AND APPLICATIONS, Lecture Notes in Computer Science, 2012, Volume 7183/2012, 433-444, DOI: 10.1007/978-3-642-28332-1_37; http://www.springerlink.com/content/t532567254652u24/. ArXiv 1011.3930. - _N. J. A. Sloane_, Jul 07 2012
%D 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 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 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D S. Vajda, Fibonacci and Lucas numbers and the Golden Section, Ellis Horwood Ltd., Chichester, 1989.
%H N. J. A. Sloane, <a href="/A000204/b000204.txt">The first 500 Lucas numbers: Table of n, L(n) for n = 1..500</a>
%H 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 R. Jovanovic, <a href="http://milan.milanovic.org/math/Math.php?akcija=SviLukasovi">First 70 Lucas numbers</a>
%H Blair Kelly, <a href="http://mersennus.net/fibonacci//lucas.txt">Factorizations of Lucas numbers</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H 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 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 _Simon 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 _Simon 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 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 José L. Ramírez, Gustavo N. Rubiano, and Rodrigo de Castro, <a href="http://arxiv.org/abs/1212.1368">A Generalization of the Fibonacci Word Fractal and the Fibonacci Snowflake</a>, arXiv preprint arXiv:1212.1368, 2012
%H N. J. A. Sloane, <a href="/A000204/a000204.html">Illustration of initial terms: the Lucas tree</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LucasNumber.html">Lucas Number</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Lucasn-StepNumber.html">Lucas n-Step Number</a>
%H <a href="/index/Cor#core">Index entries for "core" sequences</a>
%H <a href="/index/Rec#order_02">Index to sequences with linear recurrences with constant coefficients</a>, signature (1,1).
%F Expansion of x(1+2x)/(1-x-x^2). - _Simon Plouffe_, dissertation 1992; multiplied by x by R. J. Mathar, Nov 14 2007]
%F a(n) = A000045(2n)/A000045(n). - _Benoit Cloitre_, Jan 05 2003
%F For n > 1, L(n) = A000045(n+2) - A000045(n-2) (A000045 = Fibonacci numbers) - _Gerald McGarvey_, Jul 10 2004
%F 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 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 a(n)=sum{k=0..floor((n+1)/2), (n+1)*binomial(n-k+1, k)/(n-k+1)} - _Paul Barry_, Jan 30 2005
%F L(n) = A000045(n+3) - 2*A000045(n) - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Oct 07 2005
%F 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 Alberto Olivares_, Feb 28 2006
%F L(n) = A000045(n+1) + A000045(n-1). - John Blythe Dobson (j.dobson(AT)uwinnipeg.ca), Sep 29 2007
%F a(n)=2*fibonacci(n-1)+fibonacci(n), n>=1 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 05 2007
%F L(n) = term (1,1) in the 1x2 matrix [2,-1].[1,1; 1,0]^n. - _Alois P. Heinz_, Jul 25 2008
%F a(n)=phi^n+(1-phi)^n = phi^n+(-phi)^(-n) = ((1+sqrt(5))^n + (1-sqrt(5))^n)/(2^n) where phi is the Golden ratio = A001622. [From _Artur Jasinski_, Oct 05 2008]
%F a(n) = A014217(n+1)-A014217(n-1). See A153263. [From _Paul Curtz_, Dec 22 2008]
%F a(n)=((1+sqrt5)^n-(1-sqrt5)^n)/(2^n*sqrt5)+((1+sqrt5)^(n-1)-(1-sqrt5)^(n-1))/(2^(n-2)*sqrt5). [From Al Hakanson (hawkuu(AT)gmail.com), Jan 12 2009, Jan 14 2009]
%F Continued fraction for n odd: [L(n);L(n),L(n),...]=L(n)+fract(Fib(n)*phi)
%F Continued fraction for n even: [L(n);-L(n),L(n),-L(n),L(n),...]=L(n)-1+fract(Fib(n)*phi) - _Hieronymus Fischer_, October 20 2010
%F Also: [L(n)-2;1,L(n)-2,1,L(n)-2,...]=L(n)-2+fract(Fib(n)*phi) - _Hieronymus Fischer_, October 20 2010
%F INVERT transform of (1, 2, -1, -2, 1, 2,...). - Gary W. Adamson, Mar 07 2012
%F L(2n-1)=floor(phi^(2n-1)); L(2n)=ceiling(phi^(2n)). [Thomas Ordowski, Jun 15 2012]
%p A000204 := proc(n) option remember; if n <=2 then 2*n-1; else A000204(n-1)+A000204(n-2); fi; end;
%p with(combinat): A000204 := n->fibonacci(n+1)+fibonacci(n-1); # an alternative program
%p 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 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:=-z*(1+2*z)/(-1+z+z**2); [_Simon Plouffe_ in his 1992 dissertation.]
%p L:= n-> (Matrix([[2,-1]]).Matrix ([[1,1], [1,0]])^n)[1,1]; seq (L(n), n=1..50); # _Alois P. Heinz_, Jul 25 2008
%t c = (1 + Sqrt[5])/2; Table[Expand[c^n + (1-c)^n], {n, 1, 30}] (* From _Artur Jasinski_, Oct 05 2008 *)
%t Table[LucasL[n, 1], {n, 1, 36}] (* From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 09 2009 *)
%t LinearRecurrence[{1, 1},{1, 3}, 50] (* From Sture Sjöstedt, Nov 28 2011 *)
%o (PARI) A000204(n)=fibonacci(n+1)+fibonacci(n-1) [From _Michael B. Porter_, Nov 05 2009]
%o (Haskell)
%o a000204 n = a000204_list !! n
%o a000204_list = 1 : 3 : zipWith (+) a000204_list (tail a000204_list)
%o -- _Reinhard Zumkeller_, Dec 18 2011
%Y Cf. A000032, A000045, A061084, A027960, A001609, A014097, A000079, A003269, A003520, A005708, A005709, A005710, A006206, A101033, A101032, A100492, A099731, A094216, A094638, A000108.
%K core,easy,nonn,nice
%O 1,2
%A _N. J. A. Sloane_.
%E Additional comments from Yong Kong (ykong(AT)curagen.com), Dec 16 2000
%E More terms from Andre F. Labossiere (boronali(AT)laposte.net), Nov 30 2004
%E Plouffe Maple line edited by _N. J. A. Sloane_, May 13 2008
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