A132324 -id:A132324 - cp's OEIS frontend

A027871 a(n) = Product_{i=1..n} (3^i - 1).

1, 2, 16, 416, 33280, 8053760, 5863137280, 12816818094080, 84078326697164800, 1654829626053597593600, 97714379759212830706892800, 17309711516825516108403231948800

Offset: 0

N. J. A. Sloane

nonn

2*(10)^m|a(n) where 4*m <= n <= 4*m+3 for m >= 1. - G. C. Greubel, Nov 20 2015

Given probability p = 1/3^n that an outcome will occur at the n-th stage of an infinite process, then starting at n=1, 1-a(n)/A047656(n+1) is the probability that the outcome has occurred at or before the n-th iteration. The limiting ratio is 1-A100220 ~ 0.4398739. - Bob Selcoe, Mar 01 2016

G. C. Greubel, Table of n, a(n) for n = 0..50

Cf. A005329 (q=2), A027637 (q=4), A027872 (q=5), A027873 (q=6), A027875 (q=7), A027876 (q=8), A027877 (q=9), A027878 (q=10), A027879 (q=11), A027880 (q=12).

Cf. A047656, A100220, A132324.

[1] cat [&*[ 3^k-1: k in [1..n] ]: n in [1..11]]; // Vincenzo Librandi, Dec 24 2015

A027871 := proc(n)
    mul( 3^i-1,i=1..n) ;
end proc:
seq(A027871(n),n=0..8) ; # R. J. Mathar, Jul 13 2017

Table[Product[(3^k-1),{k,1,n}],{n,0,20}] (* Vaclav Kotesovec, Jul 17 2015 *)
Abs@QPochhammer[3, 3, Range[0, 10]] (* Vladimir Reshetnikov, Nov 20 2015 *)

a(n) = prod(i=1, n, 3^i-1); \\ Michel Marcus, Nov 21 2015

a(n) ~ c * 3^(n*(n+1)/2), where c = A100220 = Product_{k>=1} (1-1/3^k) = 0.560126077927948944969792243314140014379736333798... . - Vaclav Kotesovec, Nov 21 2015
a(n) = 3^(binomial(n+1,2))*(1/3;1/3){n}, where (a;q){n} is the q-Pochhammer symbol. - G. C. Greubel, Dec 24 2015
a(n) = Product_{i=1..n} A024023(i). - Michel Marcus, Dec 27 2015
G.f.: Sum_{n>=0} 3^(n*(n+1)/2)*x^n / Product_{k=0..n} (1 + 3^k*x). - Ilya Gutkovskiy, May 22 2017
From Amiram Eldar, Feb 19 2022: (Start)
Sum_{n>=0} 1/a(n) = A132324.
Sum_{n>=0} (-1)^n/a(n) = A100220. (End)

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

Original entry on oeis.org

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

Hieronymus Fischer, Aug 20 2007

Twice the constant A132324.

			3.12986803713402307587769821345767...

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.

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

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 *)

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

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

A132267 Decimal expansion of Product_{k>0} (1-1/11^k).

Original entry on oeis.org

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

Offset: 0

Hieronymus Fischer, Aug 20 2007

			0.900832706809715279949862694760...

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

Cf. A000203, A027879, A048651, A100220, A132019-A132026, A132034-A132038, A132265-A132268, A132323, A132324, A132325, A132326.

digits = 105; NProduct[1-1/11^k, {k, 1, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+3] // N[#, digits+3]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
N[QPochhammer[1/11, 1/11], 200] (* G. C. Greubel, Dec 20 2015 *)

prodinf(x=1, 1-1/11^x) \\ Altug Alkan, Dec 20 2015

Equals exp(-Sum_{n>0} sigma_1(n)/(n*11^n)) = exp(-Sum_{n>0} A000203(n)/(n*11^n)).
Equals (1/11; 1/11){infinity}, where (a;q){infinity} is the q-Pochhammer symbol. - G. C. Greubel, Dec 20 2015
From Amiram Eldar, May 09 2023: (Start)
Equals sqrt(2*Pi/log(11)) * exp(log(11)/24 - Pi^2/(6*log(11))) * Product_{k>=1} (1 - exp(-4*k*Pi^2/log(11))) (McIntosh, 1995).
Equals Sum_{n>=0} (-1)^n/A027879(n). (End)

A290000 a(n) = Product_{k=1..n-1} (3^k + 1).

Original entry on oeis.org

1, 1, 4, 40, 1120, 91840, 22408960, 16358540800, 35792487270400, 234870301468364800, 4623187014103292723200, 272999193182799435304960000, 48361261073946554365403054080000, 25701205307660304745058529866383360000, 40976048450930207702360695570691784048640000

Offset: 0

Ilya Gutkovskiy, Jun 06 2020

nonn

G. C. Greubel, Table of n, a(n) for n = 0..50

Cf. A027871, A034472, A047656, A119600, A132324, A323716.

Sequences of the form Product_{j=1..n-1} (m^j + 1): A000012 (m=0), A011782 (m=1), A028362 (m=2), this sequence (m=3), A309327 (m=4).

[n lt 3 select 1 else (&*[3^j +1: j in [1..n-1]]): n in [1..20]]; // G. C. Greubel, Feb 21 2021

Table[Product[3^k + 1, {k, 1, n - 1}], {n, 0, 14}]

a(n) = prod(k=1, n-1, 3^k + 1); \\ Michel Marcus, Jun 06 2020

from sage.combinat.q_analogues import q_pochhammer
[1]+[3^(binomial(n,2))*q_pochhammer(n-1, -1/3, 1/3) for n in (1..20)] # G. C. Greubel, Feb 21 2021

G.f. A(x) satisfies: A(x) = 1 + x * A(3*x) / (1 - x).
G.f.: Sum_{k>=0} 3^(k*(k - 1)/2) * x^k / Product_{j=0..k-1} (1 - 3^j*x).
a(0) = 1; a(n) = Sum_{k=0..n-1} 3^k * a(k).
a(n) ~ c * 3^(n*(n - 1)/2), where c = Product_{k>=1} (1 + 1/3^k) = 1.564934018567011537938849... = A132324.
a(n) = 3^(binomial(n+1,2))*(-1/3;1/3){n}, where (a;q){n} is the q-Pochhammer symbol. - G. C. Greubel, Feb 21 2021

A370466 Decimal expansion of Product_{k>=1} 1 / (1 - 1/3^k).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Mar 30 2024

			1.7853123419985341903674862960137035357...

Cf. A000244, A065446, A100220, A132324.

RealDigits[1/QPochhammer[1/3], 10, 120][[1]] (* Vaclav Kotesovec, Mar 31 2024 *)

Equals 1/QPochhammer(1/3). - Vaclav Kotesovec, Mar 31 2024

cp's OEIS Frontend

A027871 a(n) = Product_{i=1..n} (3^i - 1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A132267 Decimal expansion of Product_{k>0} (1-1/11^k).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A290000 a(n) = Product_{k=1..n-1} (3^k + 1).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A370466 Decimal expansion of Product_{k>=1} 1 / (1 - 1/3^k).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula