A319015 Decimal expansion of Sum_{k>=0} 1/2^(k^2).

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, 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, 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

Offset

1,2

Comments

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

This constant is transcendental (Nesterenko, 1996). - Amiram Eldar, Apr 30 2020

Links

Table of n, a(n) for n=1..110.

David H. Bailey, Jonathan M. Borwein, Richard E. Crandall, and Carl Pomerance, On the binary expansions of algebraic numbers, Journal de théorie des nombres de Bordeaux, Vol. 16, No. 3 (2004), pp. 487-518. See p. 490.

Yu. V. Nesterenko, Modular functions and transcendence questions (in Russian), Sbornik: Mathematics, Vol. 187 (1996), pp. 65-96, English translation, ibid., pp. 1319-1348.

Michael Ian Shamos, Property Enumerators and a Partial Sum Theorem, 2011; alternative link.

Index entries for transcendental numbers.

Formula

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

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

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

Example

1.5644684136059385793347... = (1.1001000010000001000000001...)_2.
                               | |  |    |      |        |
                               0 1  4    9     16       25

Mathematica

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

Prog

(PARI) suminf(k=0, 1/2^(k^2)) \\ Michel Marcus, Sep 08 2018

Crossrefs

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

Sequence in context: A187146 A128632 A197490 * A229481 A304490 A155591

Adjacent sequences: A319012 A319013 A319014 * A319016 A319017 A319018

Keyword

nonn,cons

Author

Ilya Gutkovskiy, Sep 07 2018

Status

approved