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%I #27 Sep 08 2022 08:46:03
%S 1,8,8,64,512,32768,16777216,549755813888,9223372036854775808,
%T 5070602400912917605986812821504,
%U 46768052394588893382517914646921056628989841375232,237142198758023568227473377297792835283496928595231875152809132048206089502588928
%N a(n) = a(n-1)*a(n-2) with a(0)=1, a(1)=8.
%C From _Peter Bala_, Nov 01 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 := 7*sum {n = 1..inf} 1/8^floor(n*phi) (= 49*sum {n = 1..inf} floor(n/phi)/8^n) = 0.89040 80325 60827 28336 ... = 1/(1 + 1/(8 + 1/(8 + 1/(64 + 1/(512 + 1/(32768 + 1/(16777216 + ...))))))). The constant c is known to be transcendental (see Adams and Davison 1977). Cf. A014565.
%C Furthermore, for k = 0,1,2,... if we define the real number X(k) = sum {n >= 1} 1/8^(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 Bruno Berselli, <a href="/A215271/b215271.txt">Table of n, a(n) for n = 0..15</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) = 8^Fibonacci(n).
%p a:= n-> 8^(<<1|1>, <1|0>>^n)[1, 2]:
%p seq(a(n), n=0..12); # _Alois P. Heinz_, Jun 17 2014
%t RecurrenceTable[{a[0] == 1, a[1] == 8, a[n] == a[n - 1] a[n - 2]}, a[n], {n, 0, 15}]
%o (Magma) [8^Fibonacci(n): n in [0..11]];
%Y Cf. A000045, A000301, A010098-A010100, A214706, A214887, A215270, A215272, A014565.
%Y Column k=8 of A244003.
%K nonn,easy
%O 0,2
%A _Bruno Berselli_, Aug 07 2012
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