A190407 Decimal expansion of Sum_{k>=1} (1/2)^A058331(k); based on a diagonal of the natural number array, A000027.

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

Offset: 0

Author

Clark Kimberling, May 10 2011

Keywords

• nonn
• cons
• easy

Comments

See A190404.

Examples

			0.12695503246504857842...
		

Links

• Danny Rorabaugh, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A058331, A190404, A190405, A190406.

Programs

Mathematica

(* See also A190404 *)
RealDigits[(EllipticTheta[3, 0, 1/4]-1)/4, 10, 120] // First (* Jean-François Alcover, Feb 13 2013 *)

PARI

th3(x)=1 + 2*suminf(n=1,x^n^2)
(th3(1/4)-1)/4 \\ Charles R Greathouse IV, Jun 06 2016

Sage

def A190407(b): # Generate the constant with b bits of precision
    return N(sum([(1/2)^(2*k^2+1) for k in range(1,b)]),b)
A190407(415) # Danny Rorabaugh, Mar 26 2015

Formula

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