login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A014551 Jacobsthal-Lucas numbers. 62

%I #263 Oct 02 2023 02:39:19

%S 2,1,5,7,17,31,65,127,257,511,1025,2047,4097,8191,16385,32767,65537,

%T 131071,262145,524287,1048577,2097151,4194305,8388607,16777217,

%U 33554431,67108865,134217727,268435457,536870911,1073741825,2147483647,4294967297,8589934591

%N Jacobsthal-Lucas numbers.

%C Also gives the number of points of period n in the subshift of finite type corresponding to the square matrix A=[1,2;1,0] (this is then given by trace(A^n)). - _Thomas Ward_, Mar 07 2001

%C Sequence is identical to its signed inverse binomial transform (autosequence of the second kind). - _Paul Curtz_, Jul 11 2008

%C a(n) can be expressed in terms of values of the Fibonacci polynomials F_n(x), computed at x=1/sqrt(2). - Tewodros Amdeberhan (tewodros(AT)math.mit.edu), Dec 15 2008

%C Pisano period lengths: 1, 1, 2, 2, 4, 2, 6, 2, 6, 4, 10, 2, 12, 6, 4, 2, 8, 6, 18, 4, ... - _R. J. Mathar_, Aug 10 2012

%C Let F(x) = Product_{n >= 0} (1 - x^(3*n+1))/(1 - x^(3*n+2)). This sequence is the simple continued fraction expansion of the real number 1 + F(-1/2) = 2.83717 78068 73232 99799 ... = 2 + 1/(1 + 1/(5 + 1/(7 + 1/(17 + ...)))). See A111317. - _Peter Bala_, Dec 26 2012

%C With different signs, 2, -1, 5, -7, 17, -31, 65, -127, 257, -511, 1025, -2047, ... is the Lucas V(-1,-2) sequence. - _R. J. Mathar_, Jan 08 2013

%C The identity 2 = 2/2 + 2^2/(2*1) - 2^3/(2*1*5) - 2^4/(2*1*5*7) + 2^5/(2*1*5*7*17) + 2^6/(2*1*5*7*17*31) - - + + can be viewed as a generalized Engel-type expansion of the number 2 to the base 2. Compare with A062510. - _Peter Bala_, Nov 13 2013

%C For n >= 2, a(n) is the number of ways to tile a 2 X n strip, where the first two columns have an extra cell at the top, with 1 X 2 dominoes and 2 X 2 squares. Shown here is one of the a(7)=127 ways for the n=7 case:

%C .___.

%C |___|_________.

%C | | | |___| |

%C |_|___|_|___|_|. - _Greg Dresden_, Sep 26 2021

%C Named by Horadam (1988) after the German mathematician Ernst Jacobsthal (1882-1965) and the French mathematician Édouard Lucas (1842-1891). - _Amiram Eldar_, Oct 02 2023

%D G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. pp. 180, 255.

%D Lind and Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, 1995. (General material on subshifts of finite type)

%D Kritkhajohn Onphaeng and Prapanpong Pongsriiam. Jacobsthal and Jacobsthal-Lucas Numbers and Sums Introduced by Jacobsthal and Tverberg. Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.6.

%D Abdelmoumène Zekiri, Farid Bencherif, Rachid Boumahdi, Generalization of an Identity of Apostol, J. Int. Seq., Vol. 21 (2018), Article 18.5.1.

%H T. D. Noe, <a href="/A014551/b014551.txt">Table of n, a(n) for n = 0..200</a>

%H Kunle Adegoke, Robert Frontczak, and Taras Goy, <a href="https://doi.org/10.7546/nntdm.2021.27.2.54-63">Partial sum of the products of the Horadam numbers with subscripts in arithmetic progression</a>, Notes on Num. Theor. and Disc. Math. (2021) Vol. 27, No. 2, 54-63.

%H Tewodros Amdeberhan, <a href="https://arxiv.org/abs/0811.4652">A note on Fibonacci-type polynomials</a>, arXiv:0811.4652 [math.NT], 2008.

%H Hacène Belbachir, Amine Belkhir, and Ihab-Eddin Djellas, <a href="https://digitalcommons.pvamu.edu/aam/vol17/iss2/15/">Permanent of Toeplitz-Hessenberg Matrices with Generalized Fibonacci and Lucas entries</a>, Applications and Applied Mathematics: An International Journal (AAM 2022), Vol. 17, Iss. 2, Art. 15, 558-570.

%H Paula Catarino, Helena Campos, and Paulo Vasco. <a href="http://ami.ektf.hu/uploads/papers/finalpdf/AMI_46_from37to53.pdf">On the Mersenne sequence</a>. Annales Mathematicae et Informaticae, 46 (2016), pp. 37-53.

%H Charles K. Cook and Michael R. Bacon, <a href="http://ami.ektf.hu/uploads/papers/finalpdf/AMI_41_from27to39.pdf">Some identities for Jacobsthal and Jacobsthal-Lucas numbers satisfying higher order recurrence relations</a>, Annales Mathematicae et Informaticae, 41 (2013), pp. 27-39.

%H Fatih Erduvan and Refik Keskin, <a href="http://www.math.nthu.edu.tw/~amen/2023/AMEN-A211023.pdf">Fibonacci And Lucas Numbers Which Are Product Of Two Jacobsal-Lucas Numbers</a> [sic], Appl. Math. E-Notes (2023) Vol. 23, 60-70.

%H M. C. Firengiz and A. Dil, <a href="http://www.nntdm.net/papers/nntdm-20/NNTDM-20-4-21-32.pdf">Generalized Euler-Seidel method for second order recurrence relations</a>, Notes on Number Theory and Discrete Mathematics, Vol. 20, 2014, No. 4, 21-32.

%H A. F. Horadam, <a href="http://www.fq.math.ca/Scanned/26-1/horadam2.pdf">Jacobsthal and Pell Curves</a>, Fib. Quart. 26, 79-83, 1988.

%H A. F. Horadam, <a href="http://www.fq.math.ca/Scanned/34-1/horadam2.pdf">Jacobsthal Representation Numbers</a>, Fib Quart. 34, 40-54, 1996.

%H D. Jhala, G. P. S. Rathore, and K. Sisodiya, <a href="http://dx.doi.org/10.12691/tjant-2-4-3">Some Properties of k-Jacobsthal Numbers with Arithmetic Indexes</a>, Turkish Journal of Analysis and Number Theory, 2014, Vol. 2, No. 4, 119-124.

%H Thomas Koshy and Ralph P. Grimaldi, <a href="https://www.fq.math.ca/Papers1/55-2/KoshyGrimaldi10272016.pdf">Ternary words and Jacobsthal numbers</a>, Fib. Quart., 55 (No. 2, 2017), 129-136.

%H Yash Puri and Thomas Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/WARD/short.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), Article 01.2.1.

%H M. Rahmani, <a href="http://dx.doi.org/10.1016/j.laa.2012.07.050">The Akiyama-Tanigawa matrix and related combinatorial identities</a>, Linear Algebra and its Applications 438 (2013) 219-230. - From _N. J. A. Sloane_, Dec 26 2012

%H Yüksel Soykan, <a href="https://doi.org/10.9734/AIR/2019/v20i230154">On Summing Formulas For Generalized Fibonacci and Gaussian Generalized Fibonacci Numbers</a>, Advances in Research (2019) Vol. 20, No. 2, 1-15, Article AIR.51824.

%H Yüksal Soykan, <a href="https://doi.org/10.9734/AJARR/2020/v8i130192">On Summing Formulas for Horadam Numbers</a>, Asian Journal of Advanced Research and Reports (2020) Vol. 8, Issue 1, 45-61.

%H Yüksel Soykan, <a href="https://doi.org/10.9734/jamcs/2020/v35i130241">Generalized Fibonacci Numbers: Sum Formulas</a>, Journal of Advances in Mathematics and Computer Science (2020) Vol. 35, No. 1, 89-104.

%H Yüksel Soykan, <a href="https://doi.org/10.9734/AJARR/2020/v9i130212">Closed Formulas for the Sums of Squares of Generalized Fibonacci Numbers</a>, Asian Journal of Advanced Research and Reports (2020) Vol. 9, No. 1, 23-39, Article no. AJARR.55441.

%H Yüksel Soykan, <a href="https://doi.org/10.9734/ACRI/2020/v20i230177">Closed Formulas for the Sums of Cubes of Generalized Fibonacci Numbers: Closed Formulas of Sum_{k=0..n} W_k^3 and Sum_{k=1..n} W_(-k)^3</a>, Archives of Current Research International (2020) Vol. 20, Issue 2, 58-69.

%H Yüksel Soykan, <a href="https://doi.org/10.34198/ejms.4220.297331">A Study on Generalized Fibonacci Numbers: Sum Formulas Sum_{k=0..n} k * x^k * W_k^3 and Sum_{k=1..n} k * x^k W_-k^3 for the Cubes of Terms</a>, Earthline Journal of Mathematical Sciences (2020) Vol. 4, No. 2, 297-331.

%H Yüksel Soykan, <a href="http://www.ijaamm.com/uploads/2/1/4/8/21481830/v8n1p1_1-14.pdf">On Generalized (r, s)-numbers</a>, Int. J. Adv. Appl. Math. and Mech. (2020) Vol. 8, No. 1, 1-14.

%H Yüksel Soykan, Erkan Taşdemir, and İnci Okumuş, <a href="https://doi.org/10.13140/RG.2.2.13499.36641">On Dual Hyperbolic Numbers With Generalized Jacobsthal Numbers Components</a>, Zonguldak Bülent Ecevit University, (Zonguldak, Turkey, 2019).

%H Anetta Szynal-Liana, Iwona Włoch, and Mirosław Liana, <a href="https://doi.org/10.17951/a.2022.76.2.33-44">Generalized commutative quaternion polynomials of the Fibonacci type</a>, Annales Math. Sect. A, Univ. Mariae Curie-Skłodowska (Poland 2022) Vol. 76, No. 2, 33-44.

%H Kai Wang, <a href="https://www.researchgate.net/publication/342010630">On Horadam Sequences and Related Infinite Series</a>, (2020).

%H Kai Wang, <a href="https://www.researchgate.net/publication/342317190">General Identities for Horadam Sequences</a>, (2020).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/JacobsthalNumber.html">Jacobsthal Number</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Lucas_sequence#Specific_names">Lucas sequence</a>.

%H OEIS Wiki, <a href="https://oeis.org/wiki/Autosequence">Autosequence</a>.

%H Volkan Yildiz, <a href="https://arxiv.org/abs/2212.08814">Some divisibility properties of Jacobsthal numbers</a>, arXiv:2212.08814 [math.CO], 2022.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,2).

%H <a href="/index/Lu#Lucas">Index entries for Lucas sequences</a>.

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.

%F a(n+1) = 2 * a(n) - (-1)^n * 3.

%F From _Len Smiley_, Dec 07 2001: (Start)

%F a(n) = 2^n + (-1)^n.

%F G.f.: (2-x)/(1-x-2*x^2). (End)

%F E.g.f.: exp(x) + exp(-2*x) produces a signed version. - _Paul Barry_, Apr 27 2003

%F a(n+1) = Sum_{k=0..floor(n/2)} binomial(n-1, 2*k)*3^(2*k)/2^(n-2). - _Paul Barry_, Feb 21 2003

%F 0, 1, 5, 7 ... is 2^n - 2*0^n + (-1)^n, the 2nd inverse binomial transform of (2^n-1)^2 (A060867). - _Paul Barry_, Sep 05 2003

%F a(n) = 2*T(n, i/(2*sqrt(2))) * (-i*sqrt(2))^n with i^2=-1. - _Paul Barry_, Nov 17 2003

%F a(n) = A078008(n) + A001045(n+1). - _Paul Barry_, Feb 12 2004

%F a(n) = 2*A001045(n+1) - A001045(n). - _Paul Barry_, Mar 22 2004

%F a(0)=2, a(1)=1, a(n) = a(n-1) + 2*a(n-2) for n > 1. - _Philippe Deléham_, Nov 07 2006

%F a(2*n+1) = Product_{d|(2*n+1)} cyclotomic(d,2). a(2^k*(2*n+1)) = Product_{d|(2*n+1)} cyclotomic(2*d,2^(2^k)). - _Miklos Kristof_, Mar 12 2007

%F a(n) = 2^{(n-1)/2}F_{n-1}(1/sqrt(2)) + 2^{(n+2)/2}F_{n-2}(1/sqrt(2)). - Tewodros Amdeberhan (tewodros(AT)math.mit.edu), Dec 15 2008

%F E.g.f.: U(0) where U(k) = 1 + (-1)^k/(2^k - 4^k*x*2/(2*x*2^k + (-1)^k*(k+1)/U(k+1))) ; (continued fraction, 3rd kind, 3-step). - _Sergei N. Gladkovskii_, Nov 02 2012

%F G.f.: U(0) where U(k) = 1 + (-1)^k/(2^k - 4^k*x*2/(2*x*2^k + (-1)^k/U(k+1))) ; (continued fraction, 3rd kind, 3-step). - _Sergei N. Gladkovskii_, Nov 02 2012

%F a(n) = sqrt(9*(A001045)^2 + (-1)^n*2^(n+2)). - _Vladimir Shevelev_, Mar 13 2013

%F G.f.: 2 + G(0)*x*(1+4*x)/(2-x), where G(k) = 1 + 1/(1 - x*(9*k-1)/( x*(9*k+8) - 2/G(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Aug 13 2013

%F a(n) = [x^n] ( (1 + x + sqrt(1 + 2*x + 9*x^2))/2 )^n for n >= 1. - _Peter Bala_, Jun 23 2015

%F For n >= 1: a(n) = A006995(2^((n+2)/2)) when n is even, a(n) = A006995(3*2^((n-1)/2) - 1) when n is odd. - _Bob Selcoe_, Sep 04 2017

%F a(n) = J(n) + 4*J(n-1), a(0)=2, where J is A001045. - _Yuchun Ji_, Apr 23 2019

%F For n >= 0, 1/(2*a(n+1)) = Sum_{m>=n} a(m)/(a(m+1)*a(m+2)). - _Kai Wang_, Mar 03 2020

%F For 4 > h >= 0, k >= 0, a(4*k+h) mod 5 = a(h) mod 5. - _Kai Wang_, May 06 2020

%F From _Kai Wang_, May 30 2020: (Start)

%F (2 - a(n+1)/a(n))/9 = Sum_{m>=n} (-2)^m/(a(m)*a(m+1)).

%F a(n) = 2*A001045(n+1) - A001045(n).

%F a(n)^2 = a(2*n) + 2*(-2)^n.

%F a(n)^2 = 9*A001045(n)^2 + 4*(-2)^n.

%F a(2*n) = 9*A001045(n)^2 + 2*(-2)^n.

%F 2*A001045(m+n) = A001045(m)*a(n) + a(m)*A001045(n).

%F 2*(-2)^n*A001045(m-n) = A001045(m)*a(n) - a(m)*A001045(n).

%F A001045(m+n) + (-2)^n*A001045(m-n) = A001045(m)*a(n).

%F A001045(m+n) - (-2)^n*A001045(m-n) = a(m)*A001045(n).

%F 2*a(m+n) = 9*A001045(m)*A001045(n) + a(m)*a(n).

%F 2*(-2)^n*a(m-n) = a(m)*a(n) - 9*A001045(m)*A001045(n).

%F a(m+n) - (-2)^n*a(m-n) = 9*A001045(m)*A001045(n).

%F a(m+n) + (-2)^n*a(m-n) = a(m)*a(n).

%F a(m+n)*a(m-n) - a(m)*a(m) = 9*(-2)^(m-n)*A001045(n)^2.

%F a(m+1)*a(n) - a(m)*a(n+1) = 9*(-2)^n*A001045(m-n). (End)

%F a(n) = F(n+1) + F(n-1) + Sum_{k=0..(n-2)} a(k)*F(n-1-k) for F(n) the Fibonacci numbers and for n > 1. - _Greg Dresden_, Jun 03 2020

%t f[n_]:=2/(n+1);x=4;Table[x=f[x];Denominator[x],{n,0,5!}] (* _Vladimir Joseph Stephan Orlovsky_, Mar 12 2010 *)

%t nxt[{n_,a_}]:={n+1,2a-3(-1)^(n+1)}; Transpose[NestList[nxt,{1,2},40]] [[2]] (* _Harvey P. Dale_, May 27 2013 *)

%t LinearRecurrence[{1, 2}, {2, 1}, 40] (* _Jean-François Alcover_, Jan 07 2019 *)

%o (Sage) [lucas_number2(n,1,-2) for n in range(0, 32)] # _Zerinvary Lajos_, Apr 30 2009

%o (PARI) a(n)=2^n+(-1)^n \\ _Charles R Greathouse IV_, Nov 20 2012

%o (Haskell)

%o a014551 n = a000079 n + a033999 n

%o a014551_list = map fst $ iterate (\(x,s) -> (2 * x - 3 * s, -s)) (2, 1)

%o -- _Reinhard Zumkeller_, Jan 02 2013

%o (Magma) [2^n + (-1)^n: n in [0..30]]; // _G. C. Greubel_, Dec 17 2017

%Y Cf. A001045 (companion "autosequence"), A019322, A066845, A111317.

%Y Cf. A135440 (first differences), A166920 (partial sums).

%Y Cf. A000079, A033999, A102345, A105723.

%Y Cf. A006995.

%K nonn,nice,easy

%O 0,1

%A _Eric W. Weisstein_

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 29 02:13 EDT 2024. Contains 371264 sequences. (Running on oeis4.)