A132038 Decimal expansion of Product_{k>0} (1-1/10^k).

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, 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, 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

Offset

0,1

Links

G. C. Greubel, Table of n, a(n) for n = 0..1500

Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge University Press, 2009, page 49.

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

Equals exp( -Sum_{n>0} sigma_1(n)/(n*10^n) ).

Equals (1/10; 1/10)_{infinity}, where (a; q)_{infinity} is the q-Pochhammer symbol. - G. C. Greubel, Nov 30 2015

From Amiram Eldar, May 09 2023: (Start)

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).

Equals Sum_{n>=0} (-1)^n/A027878(n). (End)

Example

0.8900100999989990000001000...

Mathematica

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 *)
RealDigits[QPochhammer[1/10], 10, 105][[1]] (* Jean-François Alcover, Nov 18 2015 *)
N[QPochhammer[1/10, 1/10]] (* G. C. Greubel, Nov 30 2015 *)

Prog

(PARI) prodinf(x=1, -.1^x, 1) \\ Charles R Greathouse IV, Nov 16 2013

Crossrefs

Cf. A000203, A027878, A048651, A067080, A098844, A100220, A132019, A132026.

Sequence in context: A078247 A273555 A213153 * A087495 A111253 A021533

Adjacent sequences: A132035 A132036 A132037 * A132039 A132040 A132041

Keyword

nonn,cons

Author

Hieronymus Fischer, Aug 14 2007

Status

approved