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%I #49 Sep 29 2022 11:41:13
%S 1,3,3,9,27,243,6561,1594323,10460353203,16677181699666569,
%T 174449211009120179071170507,
%U 2909321189362570808630465826492242446680483,507528786056415600719754159741696356908742250191663887263627442114881
%N a(n) = a(n-1)*a(n-2) with a(0)=1, a(1)=3.
%C From _Peter Bala_, Nov 01 2013: (Start)
%C Let phi = (1/2)*(1 + sqrt(5)) denote the golden ratio A001622. This sequence gives the simple continued fraction expansion of the constant c := 2*Sum_{n>=1} 1/3^floor(n*phi) (= 4*Sum_{n>=1} floor(n/phi)/3^n) = 0.768597560593155198508 ... = 1/(1 + 1/(3 + 1/(3 + 1/(9 + 1/(27 + 1/(243 + 1/(6561 + ...))))))). The constant c is known to be transcendental (see Adams and Davison 1977). Cf. A014565.
%C Furthermore, for k = 0,1,2,... if we put X(k) = sum {n >= 1} 1/3^(n*Fibonacci(k) + Fibonacci(k+1)*floor(n*phi)) then the real number X(k+1)/X(k) has the simple continued fraction expansion [0; a(k+1), a(k+2), a(k+3), ...] (apply Bowman 1988, Corollary 1). (End)
%H Vincenzo Librandi, <a href="/A010098/b010098.txt">Table of n, a(n) for n = 0..17</a>
%H W. W. Adams and J. L. Davison, <a href="http://www.jstor.org/stable/2041889">A remarkable class of continued fractions</a>, Proc. Amer. Math. Soc. 65 (1977), 194-198.
%H P. G. Anderson, T. C. Brown, P. J.-S. Shiue, <a href="http://people.math.sfu.ca/~vjungic/tbrown/tom-28.pdf">A simple proof of a remarkable continued fraction identity</a>, Proc. Amer. Math. Soc. 123 (1995), 2005-2009.
%H D. Bowman, <a href="http://www.fq.math.ca/26-1.html">A new generalization of Davison's theorem</a>, Fib. Quart. Volume 26 (1988), 40-45
%F a(n) = 3^Fibonacci(n).
%F a(n+1) = A000304(n+3) / A000301(n). - _Reinhard Zumkeller_, Jul 06 2014
%p a[ -1]:=1:a[0]:=3: a[1]:=3: for n from 2 to 13 do a[n]:=a[n-1]*a[n-2] od: seq(a[n], n=-1..10); # _Zerinvary Lajos_, Mar 19 2009
%t 3^Fibonacci[Range[0,13]] (* or *) t={1,3};Do[AppendTo[t,t[[-1]]*t[[-2]]],{n,12}];t (* _Vladimir Joseph Stephan Orlovsky_, Jan 21 2012 *)
%t RecurrenceTable[{a[0]==1,a[1]==3,a[n]==a[n-1]a[n-2]},a,{n,15}] (* _Harvey P. Dale_, Jan 21 2021 *)
%o (Haskell)a010098 n = a010098_list !! n
%o a010098_list = 1 : 3 : zipWith (*) a010098_list (tail a010098_list)
%o -- _Reinhard Zumkeller_, Jul 06 2014
%Y Cf. A000045, A000301, A000304, A010099, A010100, A014565, A214706, A214887, A215270, A215271, A215272.
%Y Column k=3 of A244003.
%K nonn
%O 0,2
%A _N. J. A. Sloane_
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