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

%I A190406 #22 Mar 16 2020 09:29:16
%S A190406 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,
%T A190406 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,
%U A190406 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
%N A190406 Decimal expansion of Sum_{k>=1} (1/2)^S(k-1), where S=A001844 (centered square numbers).
%C A190406 See A190404.
%H A190406 Danny Rorabaugh, <a href="/A190406/b190406.txt">Table of n, a(n) for n = 0..10000</a>
%F A190406 a(n) = floor(10^(n+1)*Sum_{j>=0} (1/2)^(2*j*(j+1)+1)) mod 10. - _Danny Rorabaugh_, Mar 26 2015
%p A190406 evalf(JacobiTheta2(0,1/4)/2^(3/2)) ; # _R. J. Mathar_, Jul 15 2013
%t A190406 (See A190404.)
%t A190406 (* or *) RealDigits[EllipticTheta[2, 0, 1/4]/(2*Sqrt[2]), 10, 120] // First (* _Jean-François Alcover_, Feb 12 2013 *)
%o A190406 (Sage)
%o A190406 def A190406(b): # Generate the constant with b bits of precision
%o A190406     return N(sum([(1/2)^(2*j*(j+1)+1) for j in range(0,b)]),b)
%o A190406 A190406(409) # _Danny Rorabaugh_, Mar 26 2015
%o A190406 (PARI) th2(x)=x^.25 + 2*suminf(n=1,x^(n+1/2)^2)
%o A190406 th2(1/4)/sqrt(8) \\ _Charles R Greathouse IV_, Jun 06 2016
%Y A190406 Cf. A190404-A190407.
%K A190406 nonn,cons,easy
%O A190406 0,1
%A A190406 _Clark Kimberling_, May 10 2011