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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).

%I A000301 #98 Jan 03 2025 12:22:38
%S A000301 1,2,2,4,8,32,256,8192,2097152,17179869184,36028797018963968,
%T A000301 618970019642690137449562112,
%U A000301 22300745198530623141535718272648361505980416,13803492693581127574869511724554050904902217944340773110325048447598592
%N A000301 a(n) = a(n-1)*a(n-2) with a(0) = 1, a(1) = 2; also a(n) = 2^Fibonacci(n).
%C A000301 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 A000301 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 A000301 Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002, p. 913.
%H A000301 T. D. Noe, <a href="/A000301/b000301.txt">Table of n, a(n) for n = 0..18</a>
%H A000301 Manosij Ghosh Dastidar and Michael Wallner, <a href="https://arxiv.org/abs/2406.16404">Bijections between Variants of Dyck Paths and Integer Compositions</a>, arXiv:2406.16404 [math.CO], 2024. See p. 1.
%H A000301 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 A000301 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 A000301 Bertrand Teguia Tabuguia, <a href="https://arxiv.org/abs/2412.20630">Computing with D-Algebraic Sequences</a>, arXiv:2412.20630 [math.AG], 2024. See p. 9.
%H A000301 <a href="/index/Di#divseq">Index to divisibility sequences</a>
%F A000301 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 A000301 a(n) = A000304(n+3) / A010098(n+1). - _Reinhard Zumkeller_, Jul 06 2014
%F A000301 Sum_{n>=0} 1/a(n) = A124091. - _Amiram Eldar_, Oct 27 2020
%F A000301 Limit_{n->oo} a(n)/a(n-1)^phi = 1. - _Peter Woodward_, Nov 24 2023
%p A000301 A000301 := proc(n) option remember;
%p A000301              if n < 2 then 1+n
%p A000301            else A000301(n-1)*A000301(n-2)
%p A000301              fi
%p A000301            end:
%p A000301 seq(A000301(n), n=0..15);
%t A000301 2^Fibonacci[Range[0, 14]] (* _Alonso del Arte_, Jul 28 2016 *)
%o A000301 (Magma) [2^Fibonacci(n): n in [0..20]]; // _Vincenzo Librandi_, Apr 18 2011
%o A000301 (PARI) a(n)=1<<fibonacci(n) \\ _Charles R Greathouse IV_, Jan 12 2012
%o A000301 (Haskell)
%o A000301 a000301 = a000079 . a000045
%o A000301 a000301_list = 1 : scanl (*) 2 a000301_list
%o A000301 -- _Reinhard Zumkeller_, Mar 20 2013
%o A000301 (SageMath) [2^fibonacci(n) for n in range(15)] # _G. C. Greubel_, Jul 29 2024
%Y A000301 Cf. A000045, A000079, A000304, A010098, A010099, A010100, A073115, A124091.
%Y A000301 Column k = 2 of A244003.
%K A000301 nonn,easy
%O A000301 0,2
%A A000301 _N. J. A. Sloane_, Mar 15 1996
%E A000301 Offset changed from 1 to 0 by _Vincenzo Librandi_, Apr 18 2011