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A132324
Decimal expansion of Product_{k>=1} (1+1/3^k).
6
1, 5, 6, 4, 9, 3, 4, 0, 1, 8, 5, 6, 7, 0, 1, 1, 5, 3, 7, 9, 3, 8, 8, 4, 9, 1, 0, 6, 7, 2, 8, 8, 3, 5, 4, 1, 6, 5, 6, 9, 4, 2, 5, 9, 1, 9, 8, 9, 5, 0, 3, 5, 0, 0, 9, 4, 9, 6, 7, 2, 1, 0, 2, 9, 9, 2, 3, 0, 2, 1, 1, 0, 7, 2, 5, 8, 0, 9, 6, 7, 6, 6, 9, 3, 9, 0, 3, 6, 6, 0, 3, 6, 7, 7, 2, 9, 6, 3, 8, 8, 1, 5, 2, 6, 0
OFFSET
1,2
COMMENTS
Half the constant A132323.
LINKS
Richard J. McIntosh, Some Asymptotic Formulae for q-Hypergeometric Series, Journal of the London Mathematical Society, Vol. 51, No. 1 (1995), pp. 120-136; alternative link.
FORMULA
(1/2)*lim sup Product{k=0..floor(log_3(n))} (1+1/floor(n/3^k)) for n-->oo.
(1/2)*lim sup A132327(n)/A132027(n) for n-->oo.
(1/2)*lim sup A132327(n)/n^((1+log_3(n))/2) for n-->oo.
(1/2)*lim sup A132328(n)/n^((log_3(n)-1)/2) for n-->oo.
exp(Sum_{n>0} 3^(-n)*Sum_{k|n} -(-1)^k/k) = exp(Sum_{n>0} A000593(n)/(n*3^n)).
(1/2)*lim sup A132327(n+1)/A132327(n) = 1.56493401856701153793884910... for n-->oo.
Equals (-1/3; 1/3)_{infinity}, where (a;q)_{infinity} is the q-Pochhammer symbol. - G. C. Greubel, Dec 01 2015
From Amiram Eldar, Feb 19 2022: (Start)
Equals (sqrt(2)/2) * exp(log(3)/24 + Pi^2/(12*log(3))) * Product_{k>=1} (1 - exp(-2*(2*k-1)*Pi^2/log(3))) (McIntosh, 1995).
Equals Sum_{n>=0} 1/A027871(n). (End)
EXAMPLE
1.56493401856701153793884910...
MATHEMATICA
digits = 105; NProduct[1+1/3^k, {k, 1, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+5] // N[#, digits+5]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
N[QPochhammer[-1/3, 1/3]] (* G. C. Greubel, Dec 01 2015 *)
KEYWORD
nonn,cons
AUTHOR
Hieronymus Fischer, Aug 20 2007
STATUS
approved