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A214887 a(n) = a(n-1)*a(n-2) with a(0)=1, a(1)=7.

%I A214887 #36 Jan 05 2025 19:51:39
%S A214887 1,7,7,49,343,16807,5764801,96889010407,558545864083284007,
%T A214887 54116956037952111668959660849,
%U A214887 30226801971775055948247051683954096612865741943
%N A214887 a(n) = a(n-1)*a(n-2) with a(0)=1, a(1)=7.
%C A214887 a(17) has 1350 digits.
%C A214887 From _Peter Bala_, Nov 01 2013: (Start)
%C A214887 Let phi = 1/2*(1 + sqrt(5)) denote the golden ratio A001622. This sequence is the simple continued fraction expansion of the constant c := 6*sum {n = 1..inf} 1/7^floor(n*phi) (= 36*sum {n = 1..inf} floor(n/phi)/7^n) = 0.87718 67194 00499 51922 ... = 1/(1 + 1/(7 + 1/(7 + 1/(49 + 1/(343 + 1/(16807 + 1/(5764801 + ...))))))). The constant c is known to be transcendental (see Adams and Davison 1977). Cf. A014565.
%C A214887 Furthermore, for k = 0,1,2,... if we define the real number X(k) = sum {n >= 1} 1/7^(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 A214887 Vincenzo Librandi, <a href="/A214887/b214887.txt">Table of n, a(n) for n = 0..16</a>
%H A214887 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 A214887 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 A214887 D. Bowman, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/26-1.html">A new generalization of Davison's theorem</a>, Fib. Quart. Volume 26 (1988), 40-45
%F A214887 a(n) = 7^Fibonacci(n).
%p A214887 a:= n-> 7^(<<1|1>, <1|0>>^n)[1, 2]:
%p A214887 seq(a(n), n=0..12);  # _Alois P. Heinz_, Jun 17 2014
%t A214887 7^Fibonacci[Range[0,10]]
%t A214887 nxt[{a_,b_}]:={b,a*b}; Transpose[NestList[nxt,{1,7},10]][[1]] (* _Harvey P. Dale_, Jun 10 2014 *)
%o A214887 (Magma) [7^Fibonacci(n): n in [0..10]];
%Y A214887 Cf. A000045, A000301, A010098, A010099, A010100, A214706, A014565, A215270, A215271, A215272.
%Y A214887 Column k=7 of A244003.
%K A214887 nonn,easy
%O A214887 0,2
%A A214887 _Vincenzo Librandi_, Aug 01 2012