This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A007404 #57 Aug 03 2024 04:18:03
%S A007404 8,1,6,4,2,1,5,0,9,0,2,1,8,9,3,1,4,3,7,0,8,0,7,9,7,3,7,5,3,0,5,2,5,2,
%T A007404 2,1,7,0,3,3,1,1,3,7,5,9,2,0,5,5,2,8,0,4,3,4,1,2,1,0,9,0,3,8,4,3,0,5,
%U A007404 5,6,1,4,1,9,4,5,5,5,3,0,0,0,6,0,4,8,5,3,1,3,2,4,8,3,9,7,2,6,5,6,1,7,5,5,8
%N A007404 Decimal expansion of Sum_{n>=0} 1/2^(2^n).
%C A007404 Kempner shows that numbers of a general form (which includes this constant) are transcendental. - _Charles R Greathouse IV_, Nov 07 2017
%D A007404 M. J. Knight, An "oceans of zeros" proof that a certain non-Liouville number is transcendental, The American Mathematical Monthly, Vol. 98, No. 10 (1991), pp. 947-949.
%H A007404 Harry J. Smith, <a href="/A007404/b007404.txt">Table of n, a(n) for n = 0..20000</a>
%H A007404 Boris Adamczewski, <a href="http://www.emis.de/journals/JIS/VOL16/Adamczewski/adam6.html">The Many Faces of the Kempner Number</a>, Journal of Integer Sequences, Vol. 16 (2013), #13.2.15.
%H A007404 David H. Bailey, Jonathan M. Borwein, Richard E. Crandall, and Carl Pomerance, <a href="https://doi.org/10.5802/jtnb.457">On the Binary Expansions of Algebraic Numbers</a>, Journal de Théorie des Nombres de Bordeaux, volume 16, number 3, 2004, pages 487-518. Also <a href="https://escholarship.org/uc/item/44t5s388">LBNL-53854</a> 2003, and authors' copies <a href="http://www.davidhbailey.com/dhbpapers/algebraic.pdf">one</a>, <a href="https://math.dartmouth.edu/~carlp/PDF/algebraic13.pdf">four</a>.
%H A007404 D. H. Bailey and H. R. P. Ferguson, <a href="/A007404/a007404.pdf">Numerical results on relations between fundamental constants using a new algorithm</a>, Mathematics of Computation, Vol.53 No. 188 (1989), 649-656. (Annotated scanned copy)
%H A007404 F. R. Bernhart & N. J. A. Sloane, <a href="/A006343/a006343.pdf">Emails, April-May 1994</a>
%H A007404 Aubrey J. Kempner, <a href="http://www.jstor.org/stable/1988833">On transcendental numbers</a>, Transactions of the American Mathematical Society 17 (1916), pp. 476-482.
%H A007404 Simon Plouffe, Plouffe's Inverter, <a href="http://www.plouffe.fr/simon/constants/2n2n.txt">sum(1/2^(2^n), n=0..infinity); to 20000 digits</a>
%H A007404 Simon Plouffe, <a href="https://web.archive.org/web/20080205202451/http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap5.html">sum(1/2^(2^n), n=0..infinity) to 1024 digits</a>
%H A007404 Jeffrey Shallit, <a href="https://doi.org/10.1016/0022-314X(79)90040-4">Simple continued fractions for some irrational numbers</a>. J. Number Theory 11 (1979), no. 2, 209-217.
%H A007404 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F A007404 Equals -Sum_{k>=1} mu(2*k)/(2^k - 1) = Sum_{k>=1, k odd} mu(k)/(2^k - 1). - _Amiram Eldar_, Jun 22 2020
%e A007404 0.81642150902189314370....
%t A007404 RealDigits[ N[ Sum[1/2^(2^n), {n, 0, Infinity}], 110]] [[1]]
%o A007404 (PARI) default(realprecision, 20080); x=suminf(n=0, 1/2^(2^n)); x*=10; for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b007404.txt", n, " ", d)); \\ _Harry J. Smith_, May 07 2009
%o A007404 (PARI) suminf(k = 0, 1/(2^(2^k))) \\ _Michel Marcus_, Mar 26 2017
%o A007404 (PARI) suminf(k=0,1.>>2^k) \\ _Charles R Greathouse IV_, Nov 07 2017
%Y A007404 Cf. A007400, A078885, A078585, A078886, A078887, A078888, A078889, A078890, A036987.
%K A007404 nonn,cons
%O A007404 0,1
%A A007404 _Simon Plouffe_
%E A007404 Edited by _Robert G. Wilson v_, Dec 11 2002
%E A007404 Deleted old PARI program _Harry J. Smith_, May 20 2009