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%I #21 Feb 12 2023 18:40:33
%S 1,10,10,100,1000,100000,100000000,10000000000000,
%T 1000000000000000000000,10000000000000000000000000000000000,
%U 10000000000000000000000000000000000000000000000000000000
%N a(n) = a(n-1)*a(n-2) with a(0)=1, a(1)=10.
%C From _Peter Bala_, Nov 11 2013: (Start)
%C Let phi = 1/2*(1 + sqrt(5)) denote the golden ratio A001622. This sequence is the simple continued fraction expansion of the constant c := 9*sum {n = 1..inf} 1/10^floor(n*phi) (= 81*sum {n = 1..inf} floor(n/phi)/10^n) = 0.90990 90990 99090 99090 ... = 1/(1 + 1/(10 + 1/(10 + 1/(100 + 1/(1000 + 1/(100000 + 1/(100000000 + ...))))))). The constant c is known to be transcendental (see Adams and Davison 1977). Cf. A014565 and A005614.
%C Furthermore, for k = 0,1,2,... if we define the real number X(k) = sum {n >= 1} 1/10^(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 Alois P. Heinz, <a href="/A010100/b010100.txt">Table of n, a(n) for n = 0..16</a>
%H W. W. Adams and J. L. Davison, <a href="https://doi.org/10.1090/S0002-9939-1977-0441879-4">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/Scanned/26-1/bowman.pdf">A new generalization of Davison's theorem</a>, Fib. Quart. Volume 26 (1988), 40-45
%F a(n) = 10^Fibonacci(n).
%p a:= n-> 10^(<<1|1>, <1|0>>^n)[1, 2]:
%p seq(a(n), n=0..12); # _Alois P. Heinz_, Jun 17 2014
%t 10^Fibonacci[Range[0,10]] (* _Harvey P. Dale_, Feb 12 2023 *)
%o (PARI) a(n) = 10^fibonacci(n); \\ _Michel Marcus_, Oct 25 2017
%Y Cf. A000045, A000301, A010098, A010099. A005614, A014565, A214706, A214887, A215270, A215271, A215272.
%Y Column k=10 of A244003.
%K nonn
%O 0,2
%A _N. J. A. Sloane_.
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