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A014551 Jacobsthal-Lucas numbers. 41
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 (t.ward(AT)uea.ac.uk), Mar 07 2001

Sequence is identical to its signed inverse binomial transform. - 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.

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; http://www.nntdm.net/papers/nntdm-20/NNTDM-20-4-21-32.pdf

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

LINKS

T. D. Noe, Table of n, a(n) for n = 0..200

T. Amdeberhan, A note on Fibonacci-type polynomials

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.

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

Eric Weisstein's World of Mathematics, Jacobsthal Number

Wikipedia, Lucas sequence

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (1,2).

Index entries for Lucas sequences

FORMULA

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

a(n) = 2^n + (-1)^n. G.f.: (2-x)/(1-x-2*x^2). - Len Smiley (smiley(AT)math.uaa.alaska.edu), Dec 07 2001

E.g.f.: exp(x)+exp(-2x) produces a signed version. - Paul Barry, Apr 27 2003

a(n+1) = sum_{k=0..floor(n/2)} binomial(n-1, 2k)3^(2k)/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(2n+1) = product_{d divides 2n+1} cyclotomic(d,2). a(2^k*(2n+1)) = product_{d divides 2n+1} cyclotomic(2d,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

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]] [[2]] (* Harvey P. Dale, May 27 2013 *)

PROG

(Sage) [lucas_number2(n, 1, -2) for n in xrange(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

CROSSREFS

Cf. A001045, A019322, A066845. A111317.

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

Cf. A000079, A033999. A102345, A105723.

Sequence in context: A227048 A183946 A005297 * A175002 A088014 A193662

Adjacent sequences:  A014548 A014549 A014550 * A014552 A014553 A014554

KEYWORD

nonn,nice,easy,changed

AUTHOR

Eric W. Weisstein

EXTENSIONS

More terms from Patrick De Geest, Jun 15 1998

STATUS

approved

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Last modified August 31 06:53 EDT 2015. Contains 261232 sequences.