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%I #36 Sep 07 2022 04:05:18
%S 1,7,8,7,2,3,1,6,5,0,1,8,2,9,6,5,9,3,3,0,1,3,2,7,4,8,9,0,3,3,7,0,0,8,
%T 3,8,5,3,3,7,9,3,1,4,0,2,9,6,1,8,1,0,9,9,7,7,8,4,7,8,1,4,7,0,5,0,5,5,
%U 5,7,4,9,1,7,5,0,6,0,5,6,8,6,9,9,1,3,1,0,0,1,8,6,3,4,0,7,5,3,3,3,0,2
%N Decimal expansion of Komornik-Loreti constant.
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, pp. 438-439.
%H Pieter Allaart, and Derong Kong, <a href="https://arxiv.org/abs/2006.07927">On the smallest base in which a number has a unique expansion</a>, arXiv:2006.07927 [math.NT], 2020.
%H Jean-Paul Allouche and Michel Cosnard, <a href="http://www.jstor.org/stable/2695302">The Komornik-Loreti constant is transcendental</a>, Amer. Math. Monthly, Vol. 107, No. 5 (May, 2000), pp. 448-449, <a href="http://www.math.jussieu.fr/~allouche/bibliorecente.html">preprint</a>.
%H Vilmos Komornik and Paola Loreti, <a href="http://www.jstor.org/stable/2589246">Unique Developments in Non-Integer Bases</a>, Amer. Math. Monthly, Vol. 105, No. 7 (Aug. - Sep., 1998), pp. 636-639.
%H Vilmos Komornik and Derong Kong, <a href="https://arxiv.org/abs/1705.00473">Bases with two expansions</a>, arXiv:1705.00473 [math.NT], 2017.
%H Vilmos Komornik, Wolfgang Steiner, and Yuru Zou, <a href="https://arxiv.org/abs/2209.02373">Unique double base expansions</a>, arXiv:2209.02373 [math.NT], 2022.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Komornik-LoretiConstant.html">Komornik-Loreti Constant</a>.
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F This number q (say) is defined by 1 = Sum_{n >= 1} A010060(n)/q^n.
%F The unique positive solution of the equation Product_{k>=0} (1 - 1/q^(2^k)) = (2-q)/(q-1) (Allouche and Cosnard, 2000). - _Amiram Eldar_, Oct 22 2020
%e 1.787231650...
%t First[ RealDigits[q /. FindRoot[ Sum[ Mod[ DigitCount[n, 2, 1], 2]/q^n, {n, 1, 2000}] == 1, {q, 1.8}, WorkingPrecision -> 120], 10, 102]](* _Jean-François Alcover_, Jun 11 2012, after PARI *)
%o (PARI) solve(q=1.7,1.8,sum(n=1,2000,(subst(Pol(binary(n)),x,1)%2)/q^n)-1)
%Y The continued-fraction expansion of this number is in A080890.
%Y Cf. A010060.
%K nonn,cons,easy
%O 1,2
%A _N. J. A. Sloane_, Jun 11 2000
%E More terms from _Ralf Stephan_, Mar 30 2003
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