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 A000301 a(n) = a(n-1)*a(n-2) with a(0) = 1, a(1) = 2; also a(n) = 2^Fibonacci(n). 40

%I

%S 1,2,2,4,8,32,256,8192,2097152,17179869184,36028797018963968,

%T 618970019642690137449562112,

%U 22300745198530623141535718272648361505980416,13803492693581127574869511724554050904902217944340773110325048447598592

%N a(n) = a(n-1)*a(n-2) with a(0) = 1, a(1) = 2; also a(n) = 2^Fibonacci(n).

%C Continued fraction expansion of s = 1.709803442861291... = Sum_{k >= 0} (1/2^floor(k * phi)) where phi is the golden ratio (1 + sqrt(5))/2. - _Benoit Cloitre_, Aug 19 2002

%C The continued fraction expansion of the above constant s is [1; 1, 2, 2, 4, ...], that of the rabbit constant r = s-1 = A014565 is [0; 1, 2, 2, 4, ...]. - _M. F. Hasler_, Nov 10 2018

%D Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002, p. 913.

%H T. D. Noe, <a href="/A000301/b000301.txt">Table of n, a(n) for n = 0..18</a>

%H J. L. Davison, <a href="http://dx.doi.org/10.1090/S0002-9939-1977-0429778-5">A series and its associated continued fraction</a>, Proc. Amer. Math. Soc., 63 (1977), 29-32.

%H Samuele Giraudo, <a href="http://arxiv.org/abs/1107.3472">Intervals of balanced binary trees in the Tamari lattice</a>, arXiv preprint arXiv:1107.3472 [math.CO], 2011-2012, and <a href="https://doi.org/10.1016/j.tcs.2011.11.020">Theor Comput Sci 420 (2012) 1-27</a>.

%H <a href="/index/Di#divseq">Index to divisibility sequences</a>

%F a(n) ~ k^phi^n with k = 2^(1/sqrt(5)) = 1.3634044... and phi the golden ratio. - _Charles R Greathouse IV_, Jan 12 2012

%F a(n) = A000304(n+3) / A010098(n+1). - _Reinhard Zumkeller_, Jul 06 2014

%F Sum_{n>=0} 1/a(n) = A124091. - _Amiram Eldar_, Oct 27 2020

%p A000301 := proc(n) option remember;

%p if n < 2 then 1+n

%p else A000301(n-1)*A000301(n-2)

%p fi

%p end:

%p seq(A000301(n), n=0..15);

%t 2^Fibonacci[Range[0, 14]] (* _Alonso del Arte_, Jul 28 2016 *)

%o (MAGMA) [2^Fibonacci(n): n in [0..20]]; // _Vincenzo Librandi_, Apr 18 2011

%o (PARI) a(n)=1<<fibonacci(n) \\ _Charles R Greathouse IV_, Jan 12 2012

%o a000301 = a000079 . a000045

%o a000301_list = 1 : scanl (*) 2 a000301_list

%o -- _Reinhard Zumkeller_, Mar 20 2013

%Y Cf. A000045, A000304, A010098, A010099, A010100, A124091.

%Y Cf. A000079.

%Y Column k = 2 of A244003.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Mar 15 1996

%E Offset changed from 1 to 0 by _Vincenzo Librandi_, Apr 18 2011

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Last modified April 14 12:11 EDT 2021. Contains 342949 sequences. (Running on oeis4.)