A319016 Decimal expansion of Sum_{k>=0} 1/2^(k*(k+1)).

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

Offset

1,2

Comments

The binary expansion is the characteristic function of the oblong numbers (A005369).

The Engel expansion of this constant are the powers of 4 (A000302). - Amiram Eldar, Dec 07 2020

Links

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

Formula

Equals theta_2(1/2)/2^(3/4), where theta_2 is the Jacobi theta function.

Equals Product_{k>=1} (1 - 1/4^k)^((-1)^k). - Antonio Graciá Llorente, Oct 01 2024

Example

1.2658700952308663684189... = (1.010001000001000000010000000001...)_2.
                               |  |   |     |       |         |
                               0  2   6    12      20        30

Mathematica

RealDigits[EllipticTheta[2, 0, 1/2]/2^(3/4), 10, 110] [[1]]

Prog

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

Crossrefs

Cf. A000302, A002378, A005369, A053763, A190405, A299998, A319015.

Sequence in context: A175293 A021083 A244928 * A262096 A011043 A021380

Adjacent sequences: A319013 A319014 A319015 * A319017 A319018 A319019

Keyword

nonn,cons

Author

Ilya Gutkovskiy, Sep 07 2018

Status

approved