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A010098 a(n) = a(n-1)*a(n-2) with a(0)=1, a(1)=3. 13

%I

%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)

%C a(n+1) = A000304(n+3) / A000301(n). - _Reinhard Zumkeller_, Jul 06 2014

%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).

%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 *)

%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, 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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Last modified July 9 14:08 EDT 2020. Contains 335543 sequences. (Running on oeis4.)