1,2

See A000032 for the version beginning 2, 1, 3, 4, 7, ...

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

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.

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

Row sums of A029635 are 1,1,3,4,7,... - Paul Barry (pbarry(AT)wit.ie), Jan 30 2005

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

Contribution from Matthew Lehman (matt.comicopia(AT)gmail.com), Nov 17 2008: (Start)

In the Fibonacci sequence, F(n) = F(n-1) + F(n-2),

for every ith number, F(n+i) = A(i)*F(n) + B(i)*F(n-i),

A(i) is given by this sequence,

Also, A(i) = F(2*i-1)/F(i-1).

B(i) = (-1)^(i+1).

For every Fibonacci number, F(n+1) = F(n) + F(n-1).

For every 2nd Fibonacci number, F(n+2) = 3*F(n) - F(n-2).

For every 3rd Fibonacci number, F(n+3) = 4*F(n) + F(n-3).

For every 4th Fibonacci number, F(n+4) = 7*F(n) - F(n-4).

For every 5th Fibonacci number, F(n+5) = 11*F(n) + F(n-5).

(End)

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]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 1968, vol. 2, p. 69.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 46.

E. Di Cera and Y. Kong, Theory of multivalent binding in one and two-dimensional lattices, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124.

Leonhard Euler, Introductio in analysin infinitorum (1748), sections 216 and 229.

G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 148.

V. E. Hoggatt, Jr., Fibonacci and Lucas Numbers. Houghton, Boston, MA, 1969.

Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001.

Yash Puri and Thomas Ward, A dynamical property unique to the Lucas sequence, Fibonacci Quarterly, Volume 39, Number 5 (November 2001), pp. 398-402.

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.

S. Vajda, Fibonacci and Lucas numbers and the Golden Section, Ellis Horwood Ltd., Chichester, 1989.

N. J. A. Sloane, The first 500 Lucas numbers: Table of n, L(n) for n = 1..500

Tanya Khovanova, Recursive Sequences

G. Everest, A. J. van der Poorten, Y. Puri and T. Ward, Integer Sequences and Periodic Points, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.3

R. Jovanovic, First 70 Lucas numbers

B. Kelly, Factorizations of Lucas numbers

C. Kimberling, Matrix Transformations of Integer Sequences, J. Integer Seqs., Vol. 6, 2003.

R. D. Knott, The Lucas Numbers in Pascal's Triangle.

A. F. Labossiere, Sobalian Coefficients.

A. F. Labossiere, Miscellaneous.

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.

N. J. A. Sloane, Illustration of initial terms: the Lucas tree

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Lucas n-Step Number

Index entries for "core" sequences

Expansion of x(1+2x)/(1-x-x^2). - S. Plouffe, dissertation 1992; multiplied by x by R. J. Mathar, Nov 14 2007]

a(n) = A000045(2n)/A000045(n). - Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 05 2003

For n > 1, L(n) = A000045(n+2) - A000045(n-2) (A000045 = Fibonacci numbers) - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Jul 10 2004

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

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

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

L(n+1) = A000045(n+4) - 2*A000045(n+1) - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Oct 07 2005

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

L(n) = A000045(n+1) + A000045(n-1). - John Blythe Dobson (j.dobson(AT)uwinnipeg.ca), Sep 29 2007

a(n)=2*fibonacci(n-1)+fibonacci(n), n>=1 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 05 2007

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

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]

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]

A000204 := proc(n) option remember; if n <=2 then 2*n-1; else A000204(n-1)+A000204(n-2); fi; end;

with(combinat): A000204 := n->fibonacci(n+1)+fibonacci(n-1); # an alternative program

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);

a:=n->2*fibonacci(n-1)+fibonacci(n): seq(a(n), n=1..34); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 05 2007

A000204:=-z*(1+2*z)/(-1+z+z**2); [S. Plouffe in his 1992 dissertation.]

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

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]

Table[LucasL[n, 1], {n, 1, 36}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 09 2009]

(PARI) A000204(n)=fibonacci(n+1)+fibonacci(n-1) [From Michael Porter (michael_b_porter(AT)yahoo.com), Nov 05 2009]

Cf. A000032, A000045, A061084, A027960.

Cf. also A001609, A014097, A000079, A003269, A003520, A005708, A005709, A005710, A006206.

Cf. A101033, A101032, A100492, A099731, A094216, A094638, A000108.

Sequence in context: A147869 A100581 A093090 this_sequence A075193 A042433 A024319

Adjacent sequences: A000201 A000202 A000203 this_sequence A000205 A000206 A000207

core,easy,nonn,nice,new

N. J. A. Sloane (njas(AT)research.att.com).

Additional comments from Yong Kong (ykong(AT)curagen.com), Dec 16 2000

More terms from Andre F. Labossiere (boronali(AT)laposte.net), Nov 30 2004

Plouffe Maple line edited by N. J. A. Sloane (njas(AT)research.att.com), May 13 2008