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%I #22 May 09 2023 08:56:53
%S 8,5,9,4,0,5,9,9,4,4,0,0,7,0,2,8,6,6,2,0,0,7,5,8,5,8,0,0,6,4,4,1,8,8,
%T 9,4,9,0,9,4,8,4,9,7,9,5,8,8,0,4,0,9,1,7,7,4,2,4,6,9,8,8,5,8,3,1,0,0,
%U 1,3,2,3,0,2,2,9,0,2,3,9,6,5,5,2,3,6,8,9,6,5,3,7,4,9,8,3,5,3,1,4,1,3,9
%N Decimal expansion of Product_{k>0} (1 - 1/8^k).
%H G. C. Greubel, <a href="/A132036/b132036.txt">Table of n, a(n) for n = 0..1200</a>
%H Richard J. McIntosh, <a href="https://doi.org/10.1112/jlms/51.1.120">Some Asymptotic Formulae for q-Hypergeometric Series</a>, Journal of the London Mathematical Society, Vol. 51, No. 1 (1995), pp. 120-136; <a href="https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.993.6895&rep=rep1&type=pdf">alternative link</a>.
%F Equals exp(-Sum_{n>0} sigma_1(n)/n*(1/8)^n) where sigma_1() is A000203().
%F Equals (1/8; 1/8)_{infinity}, where (a;q)_{infinity} is the q-Pochhammer symbol. - _G. C. Greubel_, Nov 26 2015
%F From _Amiram Eldar_, May 09 2023: (Start)
%F Equals sqrt(2*Pi/(3*log(2))) * exp(log(2)/8 - Pi^2/(18*log(2))) * Product_{k>=1} (1 - exp(-4*k*Pi^2/(3*log(2)))) (McIntosh, 1995).
%F Equals Sum_{n>=0} (-1)^n/A027876(n). (End)
%e 0.8594059944007028662007585800...
%t digits = 103; NProduct[1-1/8^k, {k, 1, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+3] // N[#, digits+3]& // RealDigits[#, 10, digits]& // First (* _Jean-François Alcover_, Feb 18 2014 *)
%t N[QPochhammer[1/8,1/8]] (* _G. C. Greubel_, Nov 26 2015 *)
%o (PARI) prodinf(x=1, 1-(1/8)^x) \\ _Altug Alkan_, Dec 01 2015
%Y Cf. A000203, A027876, A048651, A067080, A098844, A100220, A132024, A132026, A132038.
%K nonn,cons
%O 0,1
%A _Hieronymus Fischer_, Aug 14 2007
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