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%I #86 Nov 28 2023 00:08:40
%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 = A073115 = 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
%F Limit_{n->oo} a(n)/a(n-1)^phi = 1. - _Peter Woodward_, Nov 24 2023
%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 (Haskell)
%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, A073115, 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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