A132323 Decimal expansion of Product_{k>=0} (1+1/3^k).

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

Offset: 1

Author

Hieronymus Fischer, Aug 20 2007

Keywords

• nonn
• cons

Comments

Twice the constant A132324.

Examples

			3.12986803713402307587769821345767...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..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.

Crossrefs

Cf. A081845, A100220, A132019-A132026, A132034-A132038, A132265-A132268, A132324-A132326, A132327, A132328, A000593.

Programs

Mathematica

digits = 105; NProduct[1+1/3^k, {k, 0, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+3] // N[#, digits+3]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
2*N[QPochhammer[-1/3,1/3]] (* G. C. Greubel, Dec 01 2015 *)

PARI

prodinf(x=0, 1+(1/3)^x) \\ Altug Alkan, Dec 03 2015

Formula

Equals lim sup_{n->oo} Product_{0<=k<=floor(log_3(n))} (1+1/floor(n/3^k)).

Equals lim sup_{n->oo} A132327(n)/A132027(n).

Equals lim sup_{n->oo} A132327(n)/n^((1+log_3(n))/2).

Equals lim sup_{n->oo} A132328(n)/n^((log_3(n)-1)/2).

Equals 2*exp(Sum_{n>0} 3^(-n) * Sum{k|n} -(-1)^k/k) = 2*exp(Sum_{n>0} A000593(n)/(n*3^n)).

Equals lim sup_{n->oo} A132327(n+1)/A132327(n).

Equals 2*(-1/3; 1/3){infinity}, where (a;q){infinity} is the q-Pochhammer symbol. - G. C. Greubel, Dec 01 2015

Equals sqrt(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). - Amiram Eldar, May 25 2023