A319015 - OEIS

%I #32 Dec 14 2024 07:24:38

%S 1,5,6,4,4,6,8,4,1,3,6,0,5,9,3,8,5,7,9,3,3,4,7,2,9,2,7,4,2,7,2,4,7,5,

%T 6,6,2,3,0,6,2,5,8,2,6,9,9,7,0,4,3,9,0,4,6,4,4,4,5,0,5,5,9,6,0,2,8,4,

%U 8,0,1,3,3,1,7,9,5,7,8,4,0,6,6,5,9,1,3,0,6,4,0,1,6,2,4,6,9,1,4,8,4,4,7,4,0,2,4,7,1,6

%N Decimal expansion of Sum_{k>=0} 1/2^(k^2).

%C The binary expansion is the characteristic function of the squares (A010052).

%C This constant is transcendental (Nesterenko, 1996). - _Amiram Eldar_, Apr 30 2020

%H 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, Vol. 16, No. 3 (2004), pp. 487-518. See p. 490.

%H Yu. V. Nesterenko, <a href="https://doi.org/10.4213/sm158">Modular functions and transcendence questions</a> (in Russian), Sbornik: Mathematics, Vol. 187 (1996), pp. 65-96, <a href="http://dx.doi.org/10.1070/SM1996v187n09ABEH000158">English translation</a>, ibid., pp. 1319-1348.

%H Michael Ian Shamos, <a href="http://euro.ecom.cmu.edu/people/faculty/mshamos/PST.pdf">Property Enumerators and a Partial Sum Theorem</a>, 2011; <a href="https://citeseerx.ist.psu.edu/pdf/e503bec3c04c3f94cb267882724dd414e143141b">alternative link</a>.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.

%F Equals (1 + theta_3(1/2))/2, where theta_3 is the Jacobi theta function.

%F Equals 1 + Sum_{k>=1} lambda(k)/(2^k - 1), where lambda is the Liouville function (A008836). - _Amiram Eldar_, Apr 30 2020

%F Equals 1 + Sum_{k>=1} floor(sqrt(k))/2^(k+1) (Shamos, 2011, p. 4). - _Amiram Eldar_, Mar 12 2024

%e 1.5644684136059385793347... = (1.1001000010000001000000001...)_2.

%e                                | |  |    |      |        |

%e                                0 1  4    9     16       25

%t RealDigits[(1 + EllipticTheta[3, 0, 1/2])/2, 10, 110] [[1]]

%o (PARI) suminf(k=0, 1/2^(k^2)) \\ _Michel Marcus_, Sep 08 2018

%Y Cf. A000290, A002416, A008836, A010052, A190405, A299998.

%K nonn,cons

%O 1,2

%A _Ilya Gutkovskiy_, Sep 07 2018