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A319015 Decimal expansion of Sum_{k>=0} 1/2^(k^2). 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 (list; constant; graph; refs; listen; history; text; internal format)
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
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.
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
Sequence in context: A187146 A128632 A197490 * A229481 A304490 A155591
KEYWORD
nonn,cons,changed
AUTHOR
Ilya Gutkovskiy, Sep 07 2018
STATUS
approved

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Last modified March 19 01:34 EDT 2024. Contains 370952 sequences. (Running on oeis4.)