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A132022 Decimal expansion of Product_{k>=0} (1 - 1/(2*6^k)).

%I A132022 #19 May 09 2023 09:59:02
%S A132022 4,5,0,7,1,2,6,2,5,2,2,6,0,3,9,1,3,0,8,3,0,0,0,0,7,8,9,5,8,3,5,2,7,1,
%T A132022 5,5,6,0,4,4,6,7,8,5,0,0,5,4,0,0,8,5,4,7,4,3,9,0,4,5,8,3,4,8,9,2,4,4,
%U A132022 0,9,6,0,7,5,4,0,6,2,9,4,0,7,8,2,4,3,5,3,4,5,3,1,8,6,0,8,9,6,2,6,9,2,7
%N A132022 Decimal expansion of Product_{k>=0} (1 - 1/(2*6^k)).
%H A132022 G. C. Greubel, <a href="/A132022/b132022.txt">Table of n, a(n) for n = 0..2000</a>
%F A132022 Equals lim inf_{n->oo} Product_{k=0..floor(log_6(n))} floor(n/6^k)*6^k/n.
%F A132022 Equals lim inf_{n->oo} A132030(n)/n^(1+floor(log_6(n)))*6^(1/2*(1+floor(log_6(n)))*floor(log_6(n))).
%F A132022 Equals lim inf_{n->oo} A132030(n)/n^(1+floor(log_6(n)))*6^A000217(floor(log_6(n))).
%F A132022 Equals (1/2)*exp(-Sum_{n>0} 6^(-n)*Sum{k|n} 1/(k*2^k)).
%F A132022 Equals lim inf_{n->oo} A132030(n)/A132030(n+1).
%F A132022 Equals (1/2)*(1/12; 1/6)_{infinity}, where (a;q)_{infinity} is the q-Pochhammer symbol. - _G. C. Greubel_, Dec 20 2015
%F A132022 Equals Product_{n>=1} (1 - 1/A167747(n)). - _Amiram Eldar_, May 09 2023
%e A132022 0.45071262522603913...
%t A132022 digits = 103; NProduct[1-1/(2*6^k), {k, 0, Infinity}, NProductFactors -> 200, WorkingPrecision -> digits+5] // N[#, digits+5]& // RealDigits[#, 10, digits]& // First (* _Jean-François Alcover_, Feb 18 2014 *)
%t A132022 (1/2)*N[QPochhammer[1/12, 1/6], 200] (* _G. C. Greubel_, Dec 20 2015 *)
%o A132022 (PARI) prodinf(x=0, 1-1/(2*6^x)) \\ _Altug Alkan_, Dec 20 2015
%Y A132022 Cf. A000217, A048651, A098844, A067080, A132019, A132026, A132030, A132034, A167747.
%K A132022 nonn,cons
%O A132022 0,1
%A A132022 _Hieronymus Fischer_, Aug 14 2007