A132036 Decimal expansion of Product_{k>0} (1 - 1/8^k).

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, 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, 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

Offset

0,1

Links

G. C. Greubel, Table of n, a(n) for n = 0..1200

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*(1/8)^n) where sigma_1() is A000203().

Equals (1/8; 1/8)_{infinity}, where (a;q)_{infinity} is the q-Pochhammer symbol. - G. C. Greubel, Nov 26 2015

From Amiram Eldar, May 09 2023: (Start)

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).

Equals Sum_{n>=0} (-1)^n/A027876(n). (End)

Example

0.8594059944007028662007585800...

Mathematica

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 *)
N[QPochhammer[1/8, 1/8]] (* G. C. Greubel, Nov 26 2015 *)

Prog

(PARI) prodinf(x=1, 1-(1/8)^x) \\ Altug Alkan, Dec 01 2015

Crossrefs

Cf. A000203, A027876, A048651, A067080, A098844, A100220, A132024, A132026, A132038.

Sequence in context: A244810 A273985 A347195 * A132717 A200487 A070473

Adjacent sequences: A132033 A132034 A132035 * A132037 A132038 A132039

Keyword

nonn,cons

Author

Hieronymus Fischer, Aug 14 2007

Status

approved