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%I M0848 N1067 #359 May 05 2023 01:34:56
%S 2,3,7,18,47,123,322,843,2207,5778,15127,39603,103682,271443,710647,
%T 1860498,4870847,12752043,33385282,87403803,228826127,599074578,
%U 1568397607,4106118243,10749957122,28143753123,73681302247,192900153618,505019158607,1322157322203
%N Bisection of Lucas numbers: a(n) = L(2*n) = A000032(2*n).
%C Drop initial 2; then iterates of A050411 do not diverge for these starting values. - _David W. Wilson_
%C All nonnegative integer solutions of Pell equation a(n)^2 - 5*b(n)^2 = +4 together with b(n)=A001906(n), n>=0. - _Wolfdieter Lang_, Aug 31 2004
%C a(n+1) = B^(n)AB(1), n>=0, with compositions of Wythoff's complementary A(n):=A000201(n) and B(n)=A001950(n) sequences. See the W. Lang link under A135817 for the Wythoff representation of numbers (with A as 1 and B as 0 and the argument 1 omitted). E.g., 3=`10`, 7=`010`, 18=`0010`, 47=`00010`, ..., in Wythoff code. a(0) = 2 = B(1) in Wythoff code.
%C Output of Tesler's formula (as well as that of Lu and Wu) for the number of perfect matchings of an m X n Möbius band where m and n are both even specializes to this sequence for m=2. - _Sarah-Marie Belcastro_, Jul 04 2009
%C Numbers having two 1's in their base-phi representation. - _Robert G. Wilson v_, Sep 13 2010
%C Pisano period lengths: 1, 3, 4, 3, 2, 12, 8, 6, 12, 6, 5, 12, 14, 24, 4, 12, 18, 12, 9, 6, ... - _R. J. Mathar_, Aug 10 2012
%C From _Wolfdieter Lang_, Feb 18 2013: (Start)
%C a(n) is also one half of the total number of round trips, each of length 2*n, on the graph P_4 (o-o-o-o) (the simple path with 4 points (vertices) and 3 lines (or edges)). See the array and triangle A198632 for the general case of the graph P_N (there N is n and the length is l=2*k).
%C O.g.f. for w(4,l) (with zeros for odd l): y*(d/dy)S(4,y)/S(4,y) with y=1/x and Chebyshev S-polynomials (coefficients A049310). See also A198632 for a rewritten form. One half of this o.g.f. for x -> sqrt(x) produces the g.f. (2-3x)/(1-3x+x^2) given below. (End)
%C Solutions (x, y) = (a(n), a(n+1)) satisfying x^2 + y^2 = 3xy - 5. - _Michel Lagneau_, Feb 01 2014
%C Except for the first term, positive values of x (or y) satisfying x^2 - 7xy + y^2 + 45 = 0. - _Colin Barker_, Feb 16 2014
%C Except for the first term, positive values of x (or y) satisfying x^2 - 18xy + y^2 + 320 = 0. - _Colin Barker_, Feb 16 2014
%C a(n) are the numbers such that a(n)^2-2 are Lucas numbers. - _Michel Lagneau_, Jul 22 2014
%C All sequences of this form, b(n+1) = 3*b(n) - b(n-1), regardless of initial values, which includes this sequence, yield this sequence as follows: a(n) = (b(j+n) + b(j-n))/b(j), for any j, except where b(j) = 0. Also note formula below relating this a(n) to all sequences of the form G(n+1) = G(n) + G(n-1). - _Richard R. Forberg_, Nov 18 2014
%C A non-simple continued fraction expansion for F(2n*(k+1))/F(2nk) k>=1 is a(n) + (-1)/(a(n) + (-1)/(a(n) + ... + (-1)/a(n))) where a(n) appears exactly k times (F(n) denotes the n-th Fibonacci number). E.g., F(16)/F(12) equals 7 + (-1)/(7 + (-1)/7). Furthermore, these a(n) are exactly the positive integers k such that the non-simple infinite continued fraction k + (-1)/(k + (-1)/(k + (-1)/(k + ...))) belongs to Q(sqrt(5)). Compare to Benoit Cloitre and Thomas Baruchel's comments at A002878. - _Greg Dresden_, Aug 13 2019
%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 Richard P. Stanley, Enumerative combinatorics, Vol. 2. Volume 62 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1999.
%H Vincenzo Librandi, <a href="/A005248/b005248.txt">Table of n, a(n) for n = 0..1000</a>
%H Richard André-Jeannin, <a href="https://www.fq.math.ca/Scanned/29-3/andre-jeannin1.pdf">Summation of Certain Reciprocal Series Related to Fibonacci and Lucas Numbers</a>, The Fibonacci Quarterly, Vol. 29, No. 3 (1991), pp. 200-204.
%H Peter Bala, <a href="/A174500/a174500_2.pdf">Some simple continued fraction expansions for an infinite product, Part 1</a>.
%H Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Nemeth/nemeth7.html">Ellipse Chains and Associated Sequences</a>, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
%H Pooja Bhadouria, Deepika Jhala and Bijendra Singh, <a href="http://dx.doi.org/10.22436/jmcs.08.01.07">Binomial Transforms of the k-Lucas Sequences and its Properties</a>, The Journal of Mathematics and Computer Science (JMCS), Vol. 8, No. 1 (2014), pp. 81-92; sequences B_1, T_1 and R_1.
%H Noureddine Chair, <a href="https://doi.org/10.1016/j.aop.2012.09.002">Exact two-point resistance, and the simple random walk on the complete graph minus N edges</a>, Ann. Phys., Vol. 327, No. 12 (2012), pp. 3116-3129, Eq. (18).
%H Tony Crilly, <a href="http://www.jstor.org/stable/3617570">Double sequences of positive integers</a>, Math. Gaz., Vol. 69 (1985), pp. 263-271.
%H Pedro P. B. de Oliveira, Enrico Formenti, Kévin Perrot, Sara Riva and Eurico L. P. Ruivo, <a href="https://arxiv.org/abs/2004.07128">Non-maximal sensitivity to synchronism in periodic elementary cellular automata: exact asymptotic measures</a>, arXiv:2004.07128 [cs.FL], 2020.
%H Robert G. Donnelly, Molly W. Dunkum, Murray L. Huber and Lee Knupp, <a href="https://arxiv.org/abs/2012.14993">Sign-alternating Gibonacci polynomials</a>, arXiv:2012.14993 [math.CO], 2020.
%H Sergio Falcon, <a href="http://dx.doi.org/10.4236/am.2014.515216">Relationships between Some k-Fibonacci Sequences</a>, Applied Mathematics, Vol. 5, No. 15 (2014), pp. 2226-2234.
%H Margherita Maria Ferrari and Norma Zagaglia Salvi, <a href="https://www.emis.de/journals/JIS/VOL20/Salvi/salvi3.html">Aperiodic Compositions and Classical Integer Sequences</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.8.
%H Dale Gerdemann, <a href="https://www.youtube.com/watch?vEQYQ4bEUX34">Collision of Digits</a> "Also interesting are the two bisections of the Lucas numbers A005248 (digit minimizer) and A002878 (digit maximizer). I particularly like the multiples of A005248 because I have this image of the two digits piling up on top of each other and then spreading out like waves".
%H André Gougenheim, About the linear sequence of integers such that each term is the sum of the two preceding <a href="http://www.fq.math.ca/Scanned/9-3/gougenheim-a.pdf">Part 1</a> <a href="http://www.fq.math.ca/Scanned/9-3/gougenheim-b.pdf">Part 2</a>, Fib. Quart., Vol. 9, No. 3 (1971), pp. 277-295, 298.
%H Richard K. Guy, <a href="/A005246/a005246.pdf">Letter to N. J. A. Sloane</a>, Feb 1986.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>.
%H Emrah Kılıç, Yücel Türker Ulutaş and Neşe Ömür, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Omur/omur6.html">A Formula for the Generating Functions of Powers of Horadam's Sequence with Two Additional Parameters</a>, J. Int. Seq., Vol. 14 (2011), Article 11.5.6, Table 2.
%H Wentao T. Lu and F. Y. Wu, <a href="http://dx.doi.org/10.1016/S0375-9601(02)00019-1">Close-packed dimers on nonorientable surfaces</a>, Physics Letters A, Vol. 293, No. 5-6 (2002), pp. 235-246. [From Sarah-Marie Belcastro, Jul 04 2009]
%H Yun-Tak Oh, Hosho Katsura, Hyun-Yong Lee and Jung Hoon Han, <a href="https://arxiv.org/abs/1709.01344">Proposal of a spin-one chain model with competing dimer and trimer interactions</a>, arXiv:1709.01344 [cond-mat.str-el], 2017.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992, arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H Sara Riva, <a href="https://theses.hal.science/tel-03937258/document">Factorisation of Discrete Dynamical Systems</a>, Ph.D. Thesis, Univ. Côte d'Azur (France 2023).
%H Jeffrey Shallit, <a href="http://www.jstor.org/stable/2690344">An interesting continued fraction</a>, Math. Mag., Vol. 48, No. 4 (1975), pp. 207-211.
%H Jeffrey Shallit, <a href="/A005248/a005248_1.pdf">An interesting continued fraction</a>, Math. Mag., Vol. 48, No. 4 (1975), pp. 207-211. [Annotated scanned copy]
%H Jeffrey Shallit, <a href="https://arxiv.org/abs/2305.02672">Proving Properties of phi-Representations with the Walnut Theorem-Prover</a>, arXiv:2305.02672 [math.NT], 2023.
%H Jeffrey Shallit and N. J. A. Sloane, <a href="/A005248/a005248.pdf">Correspondence</a>, 1975.
%H Paweł J. Szabłowski, <a href="http://arxiv.org/abs/1403.0386">On moments of Cantor and related distributions</a>, arXiv preprint arXiv:1403.0386 [math.PR], 2014.
%H Glenn Tesler, <a href="http://dx.doi.org/10.1006/jctb.1999.1941">Matchings in Graphs on Non-Orientable Surfaces</a>, Journal of Combinatorial Theory, Series B, Vol. 78, No. 2 (2000), pp. 198-231.
%H Kai Wang, <a href="https://www.researchgate.net/publication/337943524_Fibonacci_Numbers_And_Trigonometric_Functions_Outline">Fibonacci Numbers And Trigonometric Functions Outline</a>, (2019).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PhiNumberSystem.html">Phi Number System</a>.
%H A. V. Zarelua, <a href="https://doi.org/10.1007/s11006-006-0090-y">On Matrix Analogs of Fermat’s Little Theorem</a>, Mathematical Notes, Vol. 79, No. 6 (2006), pp. 783-796. Translated from Matematicheskie Zametki, Vol. 79, No. 6 (2006), pp. 840-855.
%H <a href="/index/Rea#recur1">Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)</a>.
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%H <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1).
%F a(n) = Fibonacci(2*n-1) + Fibonacci(2*n+1).
%F G.f.: (2-3*x)/(1-3*x+x^2). - _Simon Plouffe_ in his 1992 dissertation.
%F a(n) = S(n, 3) - S(n-2, 3) = 2*T(n, 3/2) with S(n-1, 3) = A001906(n) and S(-2, x) = -1. U(n, x)=S(n, 2*x) and T(n, x) are Chebyshev's U- and T-polynomials.
%F a(n) = a(k)*a(n - k) - a(n - 2k) for all k, i.e., a(n) = 2*a(n) - a(n) = 3*a(n - 1) - a(n - 2) = 7*a(n - 2) - a(n - 4) = 18*a(n - 3) - a(n - 6) = 47*a(n - 4) - a(n - 8) etc., a(2n) = a(n)^2 - 2. - _Henry Bottomley_, May 08 2001
%F a(n) = A060924(n-1, 0) = 3*A001906(n) - 2*A001906(n-1), n >= 1. - _Wolfdieter Lang_, Apr 26 2001
%F a(n) ~ phi^(2*n) where phi=(1+sqrt(5))/2. - Joe Keane (jgk(AT)jgk.org), May 15 2002
%F a(0)=2, a(1)=3, a(n) = 3*a(n-1) - a(n-2) = a(-n). - _Michael Somos_, Jun 28 2003
%F a(n) = phi^(2*n) + phi^(-2*n) where phi=(sqrt(5)+1)/2, the golden ratio. E.g., a(4)=47 because phi^(8) + phi^(-8) = 47. - _Dennis P. Walsh_, Jul 24 2003
%F With interpolated zeros, trace(A^n)/4, where A is the adjacency matrix of path graph P_4. Binomial transform is then A049680. - _Paul Barry_, Apr 24 2004
%F a(n) = (floor((3+sqrt(5))^n) + 1)/2^n. - _Lekraj Beedassy_, Oct 22 2004
%F a(n) = ((3-sqrt(5))^n + (3+sqrt(5))^n)/2^n (Note: substituting the number 1 for 3 in the last equation gives A000204, substituting 5 for 3 gives A020876). - _Creighton Dement_, Apr 19 2005
%F a(n) = (1/(n+1/2))*Sum_{k=0..n} B(2k)*L(2n+1-2k)*binomial(2n+1, 2k) where B(2k) is the (2k)-th Bernoulli number. - _Benoit Cloitre_, Nov 02 2005
%F a(n) = term (1,1) in the 1 X 2 matrix [2,3] . [3,1; -1,0]^n. - _Alois P. Heinz_, Jul 31 2008
%F a(n) = 2*cosh(2*n*psi), where psi=log((1+sqrt(5))/2). - Al Hakanson, Mar 21 2009
%F From _Sarah-Marie Belcastro_, Jul 04 2009: (Start)
%F a(n) - (a(n) - F(2n))/2 - F(2n+1) = 0. (Tesler)
%F Product_{r=1..n} (1 + 4*(sin((4r-1)*Pi/(4n)))^2). (Lu/Wu) (End)
%F a(n) = Fibonacci(2n+6) mod Fibonacci(2n+2), n > 1. - _Gary Detlefs_, Nov 22 2010
%F a(n) = 5*Fibonacci(n)^2 + 2*(-1)^n. - _Gary Detlefs_, Nov 22 2010
%F a(n) = A033888(n)/A001906(n), n > 0. - _Gary Detlefs_, Dec 26 2010
%F a(n) = 2^(2*n) * Sum_{k=1..2} (cos(k*Pi/5))^(2*n). - _L. Edson Jeffery_, Jan 21 2012
%F From _Peter Bala_, Jan 04 2013: (Start)
%F Let F(x) = Product_{n>=0} (1 + x^(4*n+1))/(1 + x^(4*n+3)). Let alpha = 1/2*(3 - sqrt(5)). This sequence gives the simple continued fraction expansion of 1 + F(alpha) = 2.31829 56058 81914 31334 ... = 2 + 1/(3 + 1/(7 + 1/(18 + ...))).
%F Also F(-alpha) = 0.64985 97768 07374 32950 has the continued fraction representation 1 - 1/(3 - 1/(7 - 1/(18 - ...))) and the simple continued fraction expansion 1/(1 + 1/((3-2) + 1/(1 + 1/((7-2) + 1/(1 + 1/((18-2) + 1/(1 + ...))))))).
%F F(alpha)*F(-alpha) has the simple continued fraction expansion 1/(1 + 1/((3^2-4) + 1/(1 + 1/((7^2-4) + 1/(1 + 1/((18^2-4) + 1/(1 + ...))))))).
%F Added Oct 13 2019: 1/2 + 1/2*F(alpha)/F(-alpha) = 1.5142923542... has the simple continued fraction expansion 1 + 1/((3 - 2) + 1/(1 + 1/((18 - 2) + 1/(1 + 1/(123 - 2) + 1/(1 + ...))))). (End)
%F G.f.: (W(0)+6)/(5*x), where W(k) = 5*x*k + x - 6 + 6*x*(5*k-9)/W(k+1) (continued fraction). - _Sergei N. Gladkovskii_, Aug 19 2013
%F Sum_{n >= 1} 1/( a(n) - 5/a(n) ) = 1. Compare with A001906, A002878 and A023039. - _Peter Bala_, Nov 29 2013
%F 0 = a(n) * a(n+2) - a(n+1)^2 - 5 for all n in Z. - _Michael Somos_, Aug 24 2014
%F a(n) = (G(j+2n) + G(j-2n))/G(j), for n >= 0 and any j, positive or negative, except where G(j) = 0, and for any sequence of the form G(n+1) = G(n) + G(n-1) with any initial values for G(0), G(1), including non-integer values. G(n) includes Lucas, Fibonacci. Compare with A081067 for odd number offsets from j. - _Richard R. Forberg_, Nov 16 2014
%F a(n) = [x^n] ( (1 + 3*x + sqrt(1 + 6*x + 5*x^2))/2 )^n for n >= 1. - _Peter Bala_, Jun 23 2015
%F From _J. M. Bergot_, Oct 28 2015: (Start)
%F For n>0, a(n) = F(n-1) * L(n) + F(2*n+1) - (-1)^n with F(k) = A000045(k).
%F For n>1, a(n) = F(n+1) * L(n) + F(2*n-1) - (-1)^n.
%F For n>2, a(n) = 5*F(2*n-3) + 2*L(n-3) * L(n) + 8*(-1)^n. (End)
%F For n>1, a(n) = L(n-2)*L(n+2) -7*(-1)^n. - _J. M. Bergot_, Feb 10 2016
%F a(n) = 6*F(n-1)*L(n-1) - F(2*n-6) with F(n)=A000045(n) and L(n)=A000032(n). - _J. M. Bergot_, Apr 21 2017
%F a(n) = F(2*n) + 2*F(n-1)*L(n) with F(n)=A000045(n) and L(n)=A000032(n). - _J. M. Bergot_, May 01 2017
%F E.g.f.: exp(4*x/(1+sqrt(5))^2) + exp((1/4)*(1+sqrt(5))^2*x). - _Stefano Spezia_, Aug 13 2019
%F From _Peter Bala_, Oct 14 2019: (Start)
%F a(n) = F(2*n+2) - F(2*n-2) = A001906(n+1) - A001906(n-1).
%F a(n) = trace(M^n), where M is the 2 X 2 matrix [0, 1; 1, 1]^2 = [1, 1; 1, 2].
%F Consequently the Gauss congruences hold: a(n*p^k) = a(n*p^(k-1)) ( mod p^k ) for all prime p and positive integers n and k. See Zarelua and also Stanley (Ch. 5, Ex. 5.2(a) and its solution).
%F Sum_{n >= 1} (-1)^(n+1)/( a(n) + 1/a(n) ) = 1/5.
%F Sum_{n >= 1} (-1)^(n+1)/( a(n) + 3/(a(n) + 2/(a(n))) ) = 1/6.
%F Sum_{n >= 1} (-1)^(n+1)/( a(n) + 9/(a(n) + 4/(a(n) + 1/(a(n)))) ) = 1/9.
%F x*exp(Sum_{n >= 1} a(n)*x^/n) = x + 3*x^2 + 8*x^3 + 21*x^4 + ... is the o.g.f. for A001906. (End)
%F a(n) = n + 2 + Sum_{k=1..n-1} k*a(n-k). - _Yu Xiao_, May 30 2020
%F Sum_{n>=1} 1/a(n) = A153415. - _Amiram Eldar_, Nov 11 2020
%F Sum_{n>=0} 1/(a(n) + 3) = (2*sqrt(5) + 1)/10 (André-Jeannin, 1991). - _Amiram Eldar_, Jan 23 2022
%F a(n) = 2*cosh(2*n*arccsch(2)) = 2*cosh(2*n*asinh(1/2)). - _Peter Luschny_, May 25 2022
%F a(n) = (5/2)*(Sum_{k=-n..n} binomial(2*n, n+5*k)) - (1/2)*4^n. - _Greg Dresden_, Jan 05 2023
%e G.f. = 2 + 3*x + 7*x^2 + 18*x^3 + 47*x^4 + 123*x^5 + 322*x^6 + 843*x^7 + ... - _Michael Somos_, Aug 11 2009
%p a:= n-> (<<2|3>>. <<3|1>, <-1|0>>^n)[1$2]: seq(a(n), n=0..30); # _Alois P. Heinz_, Jul 31 2008
%p with(combinat): seq(5*fibonacci(n)^2+2*(-1)^n, n= 0..26);
%t a[0] = 2; a[1] = 3; a[n_] := 3a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 27}] (* _Robert G. Wilson v_, Jan 30 2004 *)
%t Fibonacci[1 + 2n] + 1/2 (-Fibonacci[2n] + LucasL[2n]) (* Tesler. _Sarah-Marie Belcastro_, Jul 04 2009 *)
%t LinearRecurrence[{3, -1}, {2, 3}, 50] (* _Sture Sjöstedt_, Nov 27 2011 *)
%t LucasL[Range[0,60,2]] (* _Harvey P. Dale_, Sep 30 2014 *)
%o (PARI) {a(n) = fibonacci(2*n + 1) + fibonacci(2*n - 1)}; /* _Michael Somos_, Jun 23 2002 */
%o (PARI) {a(n) = 2 * subst( poltchebi(n), x, 3/2)}; /* _Michael Somos_, Jun 28 2003 */
%o (Sage) [lucas_number2(n,3,1) for n in range(37)] # _Zerinvary Lajos_, Jun 25 2008
%o (Magma) [Lucas(2*n) : n in [0..100]]; // _Vincenzo Librandi_, Apr 14 2011
%o (Haskell)
%o a005248 n = a005248_list !! n
%o a005248_list = zipWith (+) (tail a001519_list) a001519_list
%o -- _Reinhard Zumkeller_, Jan 11 2012
%Y Cf. A000032, A002878 (odd-indexed Lucas numbers), A001906 (Chebyshev S(n-1, 3)), a(n) = sqrt(4+5*A001906(n)^2), A228842.
%Y a(n) = A005592(n)+1 = A004146(n)+2 = A065034(n)-1.
%Y First differences of A002878. Pairwise sums of A001519. First row of array A103997.
%Y Cf. A153415, A201157. Also Lucas(k*n): A000032 (k = 1), A014448 (k = 3), A056854 (k = 4), A001946 (k = 5), A087215 (k = 6), A087281 (k = 7), A087265 (k = 8), A087287 (k = 9), A065705 (k = 10), A089772 (k = 11), A089775 (k = 12).
%K nonn,easy
%O 0,1
%A _N. J. A. Sloane_
%E Additional comments from _Michael Somos_, Jun 23 2001
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