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

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

Offset: 0

Author

Clark Kimberling, May 10 2011

Keywords

• nonn
• cons
• easy

Comments

See A190404.

Links

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

Crossrefs

Cf. A190404-A190407.

Programs

Maple

evalf(JacobiTheta2(0,1/4)/2^(3/2)) ; # R. J. Mathar, Jul 15 2013

Mathematica

(See A190404.)
(* or *) RealDigits[EllipticTheta[2, 0, 1/4]/(2*Sqrt[2]), 10, 120] // First (* Jean-François Alcover, Feb 12 2013 *)

PARI

th2(x)=x^.25 + 2*suminf(n=1,x^(n+1/2)^2)
th2(1/4)/sqrt(8) \\ Charles R Greathouse IV, Jun 06 2016

Sage

def A190406(b): # Generate the constant with b bits of precision
    return N(sum([(1/2)^(2*j*(j+1)+1) for j in range(0,b)]),b)
A190406(409) # Danny Rorabaugh, Mar 26 2015

Formula

a(n) = floor(10^(n+1)*Sum_{j>=0} (1/2)^(2*j*(j+1)+1)) mod 10. - Danny Rorabaugh, Mar 26 2015