Search: id:A000204 Results 1-1 of 1 results found. %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, The first 500 Lucas numbers: Table of n, L(n) for n = 1..500 %H A000204 Tanya Khovanova, Recursive Sequences %H A000204 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 %H A000204 R. Jovanovic, First 70 Lucas numbers %H A000204 B. Kelly, Factorizations of Lucas numbers %H A000204 C. Kimberling, Matrix Transformations of Integer Sequences, J. Integer Seqs., Vol. 6, 2003. %H A000204 R. D. Knott, The Lucas Numbers in Pascal's Triangle. %H A000204 A. F. Labossiere, Sobalian Coefficients. %H A000204 A. F. Labossiere, Miscellaneous. %H A000204 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. %H A000204 S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992. %H A000204 Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1. %H A000204 N. J. A. Sloane, Illustration of initial terms: the Lucas tree %H A000204 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. %H A000204 Eric Weisstein's World of Mathematics, Lucas n-Step Number %H A000204 Index entries for "core" sequences %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 Search completed in 0.003 seconds