|
%I #34 May 09 2023 08:57:05
%S 8,9,0,0,1,0,0,9,9,9,9,8,9,9,9,0,0,0,0,0,0,1,0,0,0,0,9,9,9,9,9,9,9,9,
%T 8,9,9,9,9,9,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,9,9,9,9,9,9,9,9,9,9,9,
%U 9,8,9,9,9,9,9,9,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,9,9,9
%N Decimal expansion of Product_{k>0} (1-1/10^k).
%H G. C. Greubel, <a href="/A132038/b132038.txt">Table of n, a(n) for n = 0..1500</a>
%H Philippe Flajolet and Robert Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/AnaCombi/anacombi.html">Analytic Combinatorics</a>, Cambridge University Press, 2009, page 49.
%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*10^n) ).
%F Equals (1/10; 1/10)_{infinity}, where (a; q)_{infinity} is the q-Pochhammer symbol. - _G. C. Greubel_, Nov 30 2015
%F From _Amiram Eldar_, May 09 2023: (Start)
%F Equals sqrt(2*Pi/log(10)) * exp(log(10)/24 - Pi^2/(6*log(10))) * Product_{k>=1} (1 - exp(-4*k*Pi^2/log(10))) (McIntosh, 1995).
%F Equals Sum_{n>=0} (-1)^n/A027878(n). (End)
%e 0.8900100999989990000001000...
%t digits = 105; Clear[p]; p[n_] := p[n] = RealDigits[Product[1-1/10^k , {k, 1, n}], 10, digits] // First; p[10]; p[n=20]; While[p[n] != p[n/2], n = 2*n]; p[n] (* _Jean-François Alcover_, Feb 17 2014 *)
%t RealDigits[QPochhammer[1/10], 10, 105][[1]] (* _Jean-François Alcover_, Nov 18 2015 *)
%t N[QPochhammer[1/10,1/10]] (* _G. C. Greubel_, Nov 30 2015 *)
%o (PARI) prodinf(x=1,-.1^x,1) \\ _Charles R Greathouse IV_, Nov 16 2013
%Y Cf. A000203, A027878, A048651, A067080, A098844, A100220, A132019, A132026.
%K nonn,cons
%O 0,1
%A _Hieronymus Fischer_, Aug 14 2007
|
