A190405 - OEIS

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

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OFFSET

0,1

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)

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.

EXAMPLE

0.64163256065515386629...

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 *)

PROG

(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

(PARI) th2(x)=2*x^.25 + 2*suminf(n=1, x^(n+1/2)^2)