A132266 Decimal expansion of Product_{k>=0} (1 - 1/(2*12^k)).

4, 7, 7, 3, 5, 2, 1, 7, 0, 2, 5, 4, 8, 9, 3, 8, 0, 1, 9, 8, 3, 3, 4, 2, 8, 6, 3, 6, 5, 8, 2, 0, 2, 3, 0, 3, 5, 0, 8, 8, 5, 9, 6, 4, 2, 1, 4, 4, 4, 5, 8, 5, 0, 0, 7, 6, 0, 3, 4, 5, 6, 1, 3, 8, 9, 1, 4, 1, 2, 8, 8, 8, 5, 7, 9, 1, 6, 3, 5, 2, 4, 7, 7, 2, 8, 0, 9, 4, 1, 6, 5, 3, 5, 3, 6, 1, 1, 3, 5, 0, 0, 3, 7, 2, 5

Offset

0,1

Links

G. C. Greubel, Table of n, a(n) for n = 0..2000

Formula

lim inf (Product_{k=0..floor(log_12(n))} floor(n/12^k)*12^k/n) for n-->oo.

lim inf A132264(n)*12^((1+floor(log_12(n)))*floor(log_12(n))/2)/n^(1+floor(log_12(n))) for n-->oo.

lim inf A132264(n)*12^A000217(floor(log_12(n)))/n^(1+floor(log_12(n))) for n-->oo.

(1/2)*exp(-Sum_{n>0} 12^(-n)*Sum_{k|n} 1/(k*2^k)).

lim inf A132264(n)/A132264(n+1) = 0.47735217025489380... for n-->oo.

Equals (1/2)*(1/24; 1/12)_{infinity}, where (a;q)_{infinity} is the q-Pochhammer symbol. - G. C. Greubel, Dec 20 2015

Example

0.47735217025489380198334286365820...

Mathematica

digits = 105; NProduct[1-1/(2*12^k), {k, 0, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+5] // N[#, digits+5]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
(1/2)*N[QPochhammer[1/24, 1/12], 200] (* G. C. Greubel, Dec 20 2015 *)

Prog

(PARI) prodinf(x=0, 1-1/(2*12^x)) \\ Altug Alkan, Dec 20 2015

Crossrefs

Cf. A000217, A048651, A132019-A132026, A132034-A132038, A132265-A132268, A132323-A132326, A132264.

Sequence in context: A097916 A176529 A086775 * A084104 A271026 A093582

Adjacent sequences: A132263 A132264 A132265 * A132267 A132268 A132269

Keyword

nonn,cons

Author

Hieronymus Fischer, Aug 20 2007

Status

approved