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 A014551 Jacobsthal-Lucas numbers. 58
 2, 1, 5, 7, 17, 31, 65, 127, 257, 511, 1025, 2047, 4097, 8191, 16385, 32767, 65537, 131071, 262145, 524287, 1048577, 2097151, 4194305, 8388607, 16777217, 33554431, 67108865, 134217727, 268435457, 536870911, 1073741825, 2147483647, 4294967297, 8589934591 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS 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 Sequence is identical to its signed inverse binomial transform (autosequence of the second kind). - Paul Curtz, Jul 11 2008 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 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 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 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 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 REFERENCES G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. pp. 180, 255. T. Koshy and R. P. Grimaldi, Ternary words and Jacobsthal numbers, Fib. Quart., 55 (No. 2, 2017), 129-136. Lind and Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, 1995. (General material on subshifts of finite type) 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. Abdelmoumène Zekiri, Farid Bencherif, Rachid Boumahdi, Generalization of an Identity of Apostol, J. Int. Seq., Vol. 21 (2018), Article 18.5.1. LINKS T. D. Noe, Table of n, a(n) for n = 0..200 T. Amdeberhan, A note on Fibonacci-type polynomials, arXiv:0811.4652 [math.NT], 2008. Paula Catarino, Helena Campos, and Paulo Vasco. On the Mersenne sequence. Annales Mathematicae et Informaticae, 46 (2016) pp. 37-53. Charles K. Cook and Michael R. Bacon, Some identities for Jacobsthal and Jacobsthal-Lucas numbers satisfying higher order recurrence relations, Annales Mathematicae et Informaticae, 41 (2013) pp. 27-39. M. C. Firengiz, A. Dil, Generalized Euler-Seidel method for second order recurrence relations, Notes on Number Theory and Discrete Mathematics, Vol. 20, 2014, No. 4, 21-32. A. F. Horadam, Jacobsthal and Pell Curves, Fib. Quart. 26, 79-83, 1988. A. F. Horadam, Jacobsthal Representation Numbers, Fib Quart. 34, 40-54, 1996. D. Jhala, G. P. S. Rathore, K. Sisodiya, Some Properties of k-Jacobsthal Numbers with Arithmetic Indexes, Turkish Journal of Analysis and Number Theory, 2014, Vol. 2, No. 4, 119-124. Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1. M. Rahmani, The Akiyama-Tanigawa matrix and related combinatorial identities, Linear Algebra and its Applications 438 (2013) 219-230. - From N. J. A. Sloane, Dec 26 2012 Yüksel Soykan, On Summing Formulas For Generalized Fibonacci and Gaussian Generalized Fibonacci Numbers, Advances in Research (2019) Vol. 20, No. 2, 1-15, Article AIR.51824. Yüksal Soykan, On Summing Formulas for Horadam Numbers, Asian Journal of Advanced Research and Reports (2020) Vol. 8, Issue 1, 45-61. Yüksel Soykan, Generalized Fibonacci Numbers: Sum Formulas, Journal of Advances in Mathematics and Computer Science (2020) Vol. 35, No. 1, 89-104. Yüksel Soykan, Closed Formulas for the Sums of Squares of Generalized Fibonacci Numbers, Asian Journal of Advanced Research and Reports (2020) Vol. 9, No. 1, 23-39, Article no. AJARR.55441. Yüksel Soykan, 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, Archives of Current Research International (2020) Vol. 20, Issue 2, 58-69. Yüksel Soykan, 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, Earthline Journal of Mathematical Sciences (2020) Vol. 4, No. 2, 297-331. Yüksel Soykan, On Generalized (r, s)-numbers, Int. J. Adv. Appl. Math. and Mech. (2020) Vol. 8, No. 1, 1-14. Yüksel Soykan, Erkan Taşdemir, İnci Okumuş, On Dual Hyperbolic Numbers With Generalized Jacobsthal Numbers Components, Zonguldak Bülent Ecevit University, (Zonguldak, Turkey, 2019). Kai Wang, On Horadam Sequences and Related Infinite Series, (2020). Kai Wang, General Identities for Horadam Sequences, (2020). Eric Weisstein's World of Mathematics, Jacobsthal Number Wikipedia, Lucas sequence Index entries for linear recurrences with constant coefficients, signature (1,2). FORMULA a(n+1) = 2 * a(n) - (-1)^n * 3. From Len Smiley, Dec 07 2001: (Start) a(n) = 2^n + (-1)^n. G.f.: (2-x)/(1-x-2*x^2). (End) E.g.f.: exp(x) + exp(-2*x) produces a signed version. - Paul Barry, Apr 27 2003 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 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 a(n) = 2*T(n, i/(2*sqrt(2))) * (-i*sqrt(2))^n with i^2=-1. - Paul Barry, Nov 17 2003 a(n) = A078008(n) + A001045(n+1). - Paul Barry, Feb 12 2004 a(n) = 2*A001045(n+1) - A001045(n). - Paul Barry, Mar 22 2004 a(0)=2, a(1)=1, a(n) = a(n-1) + 2*a(n-2) for n > 1. - Philippe Deléham, Nov 07 2006 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 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 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 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 a(n) = sqrt(9*(A001045)^2 + (-1)^n*2^(n+2)). - Vladimir Shevelev, Mar 13 2013 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 a(n) = [x^n] ( (1 + x + sqrt(1 + 2*x + 9*x^2))/2 )^n for n >= 1. - Peter Bala, Jun 23 2015 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 a(n) = J(n) + 4*J(n-1), a(0)=2, where J is A001045. - Yuchun Ji, Apr 23 2019 For n >= 0, 1/(2*a(n+1)) = Sum_{m>=n} a(m)/(a(m+1)*a(m+2)). - Kai Wang, Mar 03 2020 For 4 > h >= 0, k >= 0, a(4*k+h) mod 5 = a(h) mod 5. - Kai Wang, May 06 2020 From Kai Wang, May 30 2020: (Start) (2 - a(n+1)/a(n))/9 = Sum_{m>=n} (-2)^m/(a(m)*a(m+1)). a(n) = 2*A001045(n+1) - A001045(n). a(n)^2 = a(2*n) + 2*(-2)^n. a(n)^2 = 9*A001045(n)^2 + 4*(-2)^n. a(2*n) = 9*A001045(n)^2 + 2*(-2)^n. 2*A001045(m+n) = A001045(m)*a(n) + a(m)*A001045(n). 2*(-2)^n*A001045(m-n) = A001045(m)*a(n) - a(m)*A001045(n). A001045(m+n) + (-2)^n*A001045(m-n) = A001045(m)*a(n). A001045(m+n) - (-2)^n*A001045(m-n) = a(m)*A001045(n). 2*a(m+n) = 9*A001045(m)*A001045(n) + a(m)*a(n). 2*(-2)^n*a(m-n) = a(m)*a(n) - 9*A001045(m)*A001045(n). a(m+n) - (-2)^n*a(m-n) = 9*A001045(m)*A001045(n). a(m+n) + (-2)^n*a(m-n) = a(m)*a(n). a(m+n)*a(m-n) - a(m)*a(m) = 9*(-2)^(m-n)*A001045(n)^2. a(m+1)*a(n) - a(m)*a(n+1) = 9*(-2)^n*A001045(m-n). (End) 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 MATHEMATICA f[n_]:=2/(n+1); x=4; Table[x=f[x]; Denominator[x], {n, 0, 5!}] (* Vladimir Joseph Stephan Orlovsky, Mar 12 2010 *) nxt[{n_, a_}]:={n+1, 2a-3(-1)^(n+1)}; Transpose[NestList[nxt, {1, 2}, 40]] [] (* Harvey P. Dale, May 27 2013 *) LinearRecurrence[{1, 2}, {2, 1}, 40] (* Jean-François Alcover, Jan 07 2019 *) PROG (Sage) [lucas_number2(n, 1, -2) for n in range(0, 32)] # Zerinvary Lajos, Apr 30 2009 (PARI) a(n)=2^n+(-1)^n \\ Charles R Greathouse IV, Nov 20 2012 (Haskell) a014551 n = a000079 n + a033999 n a014551_list = map fst \$ iterate (\(x, s) -> (2 * x - 3 * s, -s)) (2, 1) -- Reinhard Zumkeller, Jan 02 2013 (MAGMA) [2^n + (-1)^n: n in [0..30]]; // G. C. Greubel, Dec 17 2017 CROSSREFS Cf. A001045, A019322, A066845. A111317. Cf. A135440 (first differences), A166920 (partial sums). Cf. A000079, A033999. A102345, A105723. Cf. A006995. Sequence in context: A293719 A291377 A005297 * A175002 A088014 A193662 Adjacent sequences:  A014548 A014549 A014550 * A014552 A014553 A014554 KEYWORD nonn,nice,easy AUTHOR STATUS approved

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Last modified April 15 01:24 EDT 2021. Contains 342974 sequences. (Running on oeis4.)