|
%I #60 Dec 15 2017 17:35:06
%S 1,2,2,6,7,4,2,0,1,0,7,2,0,3,5,3,2,4,4,4,1,7,6,3,0,2,3,0,4,5,5,3,6,1,
%T 6,5,5,8,7,1,4,0,9,6,9,0,4,4,0,2,5,0,4,1,9,6,4,3,2,9,7,3,0,1,2,1,4,0,
%U 2,2,1,3,8,3,1,5,3,1,2,1,6,8,4,5,2,6,2,1,5,6,2,4,9,4,7,9,7,7,4,1,2,5,9,1,3
%N Decimal expansion of Fibonacci factorial constant.
%C The Fibonacci factorial constant is associated with the Fibonacci factorial A003266.
%C Two closely related constants are A194159 and A194160. [_Johannes W. Meijer_, Aug 21 2011]
%D S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.2.5.
%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison Wesley, 1990, pp. 478, 571.
%H Harry J. Smith, <a href="/A062073/b062073.txt">Table of n, a(n) for n=1..5000</a>
%H M. Griffiths, <a href="http://www.fq.math.ca/Papers1/46_47-3/Griffiths.pdf">Symmetric rational expressions in the Fibonacci numbers</a>, Fib. Q., 46/47 (2008/2009), 262-267. [_N. J. A. Sloane_, Dec 05 2009]
%H Simon Plouffe, <a href="http://www.plouffe.fr/simon/constants/fibofact.txt">Fibonacci factorials</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FibonacciFactorialConstant.html">Fibonacci Factorial Constant</a>
%F C = (1-a)*(1-a^2)*(1-a^3)... 1.2267420... where a = -1/phi^2 and where phi is the Golden ratio = 1/2 + sqrt(5)/2.
%F C = QPochhammer[ -1/GoldenRatio^2]. [_Eric W. Weisstein_, Dec 01 2009]
%F C = A194159 * A194160. [_Johannes W. Meijer_, Aug 21 2011]
%F C = exp( Sum_{k>=1} 1/(k*(1-(-(3+sqrt(5))/2)^k)) ). - _Vaclav Kotesovec_, Jun 08 2013
%F C = Sum_{k = -inf .. inf} (-1)^((k-1)*k/2) / phi^((3*k-1)*k), where phi = (1 + sqrt(5))/2. - _Vladimir Reshetnikov_, Sep 20 2016
%e 1.226742010720353244417630230455361655871409690440250419643297301214...
%t RealDigits[N[QPochhammer[-1/GoldenRatio^2], 105]][[1]] (* _Alonso del Arte_, Dec 20 2010 *)
%t RealDigits[N[Re[(-1)^(1/24) * GoldenRatio^(1/12) / 2^(1/3) * EllipticThetaPrime[1,0,-I/GoldenRatio]^(1/3)], 120]][[1]] (* _Vaclav Kotesovec_, Jul 19 2015, after _Eric W. Weisstein_ *)
%o (PARI) \p 1300 a=-1/(1/2+sqrt(5)/2)^2; prod(n=1,17000,(1-a^n))
%o (PARI) { default(realprecision, 5080); p=-1/(1/2 + sqrt(5)/2)^2; x=prodinf(k=1, 1-p^k); for (n=1, 5000, d=floor(x); x=(x-d)*10; write("b062073.txt", n, " ", d)) } \\ _Harry J. Smith_, Jul 31 2009
%Y Cf. A003266, A003267, A003268, A056569, A062072, A062381, A135407, A181926.
%Y Cf. A218490, A253924, A256831, A259314, A259405.
%K easy,nonn,cons
%O 1,2
%A _Jason Earls_, Jun 27 2001
|
