A190405 Decimal expansion of Sum_{k>=1} (1/2)^T(k), where T=A000217 (triangular numbers); based on column 1 of the natural number array, A000027.

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

Offset: 0

Author

Clark Kimberling, May 10 2011

Keywords

• nonn
• cons
• easy

Comments

See A190404.

Binary expansion is .1010010001... (A023531). - Rick L. Shepherd, Jan 05 2014

From Amiram Eldar, Dec 07 2020: (Start)

This constant is not a quadratic irrational (Duverney, 1995).

The Engel expansion of this constant are the powers of 2 (A000079) above 1. (End)

Examples

			0.64163256065515386629...
		

Links

• Danny Rorabaugh, Table of n, a(n) for n = 0..10000
• Daniel Duverney, Sommes de deux carrés et irrationalité de valeurs de fonctions thêta, Comptes rendus de l'Académie des sciences, Série 1, Mathématique, Vol. 320, No. 9 (1995), pp. 1041-1044.

Crossrefs

A190404: (1/2)(1 + Sum_{k>=1} (1/2)^T(k)), where T = A000217 (triangular numbers).

A190405: Sum_{k>=1} (1/2)^T(k), where T = A000217 (triangular numbers).

A190406: Sum_{k>=1} (1/2)^S(k-1), where S = A001844 (centered square numbers).

A190407: Sum_{k>=1} (1/2)^V(k), where V = A058331 (1 + 2*k^2).

Cf. A000079.

Programs

Mathematica

RealDigits[EllipticTheta[2, 0, 1/Sqrt[2]]/2^(7/8) - 1, 10, 120] // First (* Jean-François Alcover, Feb 12 2013 *)
RealDigits[Total[(1/2)^Accumulate[Range[50]]],10,120][[1]] (* Harvey P. Dale, Oct 18 2013 *)
(* See also A190404 *)

PARI

th2(x)=2*x^.25 + 2*suminf(n=1,x^(n+1/2)^2)
th2(sqrt(.5))/2^(7/8)-1 \\ Charles R Greathouse IV, Jun 06 2016

Sage

def A190405(b):  # Generate the constant with b bits of precision
    return N(sum([(1/2)^(j*(j+1)/2) for j in range(1,b)]),b)
A190405(409) # Danny Rorabaugh, Mar 25 2015