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A014551
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Jacobsthal-Lucas numbers.
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38
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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
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OFFSET
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0,1
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COMMENTS
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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). [From 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
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REFERENCES
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G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. pp. 180, 255.
Horadam, A. F. ``Jacobsthal and Pell Curves.'' Fib. Quart. 26, 79-83, 1988.
Horadam, A. F. ``Jacobsthal Representation Numbers.'' Fib Quart. 34, 40-54, 1996.
Lind and Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, 1995. (General material on subshifts of finite type)
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
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..200
T. Amdeberhan, A note on Fibonacci-type polynomials
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Eric Weisstein's World of Mathematics, Jacobsthal Number
Wikipedia, Lucas sequence
Index entries for sequences related to Chebyshev polynomials.
Index entries for sequences related to linear recurrences with constant coefficients, signature (1,2).
Index entries for Lucas sequences
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FORMULA
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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)=2T(n, i/(2sqrt(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 DELEHAM, 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)). [From 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
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MATHEMATICA
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f[n_]:=2/(n+1); x=4; Table[x=f[x]; Denominator[x], {n, 0, 5!}] [From Vladimir Joseph Stephan Orlovsky, Mar 12 2010]
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PROG
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(Sage) [lucas_number2(n, 1, -2) for n in xrange(0, 32)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 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
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CROSSREFS
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Cf. A001045, A019322, A066845. A111317.
Cf. A135440 (first differences), A166920 (partial sums); A000079, A033999.
Sequence in context: A105459 A183946 A005297 * A175002 A088014 A193662
Adjacent sequences: A014548 A014549 A014550 * A014552 A014553 A014554
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KEYWORD
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nonn,nice,easy
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AUTHOR
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Eric W. Weisstein
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EXTENSIONS
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More terms from Patrick De Geest, Jun 15 1998.
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STATUS
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approved
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