A081845 -id:A081845 - cp's OEIS frontend

A345987 Decimal expansion of constant mu(ell) arising in study of complexity of Euclidean algorithm.

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

Offset: 1

N. J. A. Sloane, Jul 12 2021

The constant is (12/Pi^2)*log(Product_{i>=0} (1+1/2^i)).

			1.89919324391088067944828320698125120791994827100906...

Lhote, Loïck, and Brigitte Vallée. "Sharp estimates for the main parameters of the Euclid Algorithm." In Latin American Symposium on Theoretical Informatics, pp. 689-702. Springer, Berlin, Heidelberg, 2006.

Cf. A081845.

evalf(12/Pi^2*log(product(1+1/2^i, i=0..infinity)), 120);  # Alois P. Heinz, Jul 12 2021

RealDigits[(12/Pi^2)*Log[Product[1 + 1/2^i, {i, 0, Infinity}]], 10, 105][[1]] (* Amiram Eldar, Jul 12 2021 *)

Equals 12*log(QPochhammer(-1,1/2))/Pi^2. - Stefano Spezia, Jul 12 2021

A371748 Decimal expansion of Product_{k>=0} (1 + 1/4^k).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 05 2024

			2.71181934772695876069108846970791860244...

Kelvin Voskuijl, Table of n, a(n) for n = 1..20000

Cf. A000302, A065445, A081845, A100221, A132323, A371747, A371749, A371751.

RealDigits[QPochhammer[-1, 1/4], 10, 100][[1]]

Equals A065445^2. - Hugo Pfoertner, Apr 05 2024

A371751 Decimal expansion of Product_{k>=0} (1 + 1/5^k).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 05 2024

			2.5210016134202150647777463154782130132...

Kelvin Voskuijl, Table of n, a(n) for n = 1..20000

Cf. A000351, A081845, A100222, A132323, A371748, A371750, A371752.

RealDigits[QPochhammer[-1, 1/5], 10, 100][[1]]
RealDigits[Times@@Table[1+1/5^k,{k,0,1000}],10,100][[1]] (* Harvey P. Dale, Sep 17 2024 *)

A005477 a(n) = 2^(n-1)(2^n - 1)Product_{j=1..n-1} (2^j + 1).

Original entry on oeis.org

0, 1, 18, 420, 16200, 1138320, 152681760, 40012315200, 20727639504000, 21349793828563200, 43852643645542617600, 179883715700853141120000, 1474687052822610564537600000, 24170122236238340825650936320000, 792151597973733707815459821941760000, 51919200227802645600849976559054284800000

Offset: 0

N. J. A. Sloane

nonn

G. C. Greubel, Table of n, a(n) for n = 0..75

Cf. A081845.

[n le 1 select n else 2^(n-1)*(2^n -1)*(&*[2^j+1: j in [1..n-1]]): n in [0..25]]; // G. C. Greubel, Nov 25 2022

f := i->2^(i-1)*(2^i-1)*product( '2^j+1','j'=1..i-1);

Table[2^(n-1) (2^n-1)Product[2^j+1,{j,n-1}],{n,0,20}] (* Harvey P. Dale, Feb 02 2022 *)
Table[2^(n-2)*(2^n-1)*QPochhammer[-1,2,n], {n,0,30}] (* G. C. Greubel, Nov 25 2022 *)

def A005477(n): return 2^(n-2)*(2^n-1)*product(2^j+1 for j in range(n))
[A005477(n) for n in range(30)] # G. C. Greubel, Nov 25 2022

a(n) = 2^(n-2)*(2^n - 1)*QPochhammer(n, -1, 2). - G. C. Greubel, Nov 25 2022

a(n) ~ c * 2^((n^2+3*n-4)/2), where c = Product_{k>=0} (1 + 1/2^k) = A081845. - Amiram Eldar, Aug 18 2025

a(0) prepended by G. C. Greubel, Nov 25 2022

A330863 Decimal expansion of Product_{k>=1} (1 + 1/(-2)^k).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Apr 28 2020

			(1 - 1/2) * (1 + 1/2^2) * (1 - 1/2^3) * (1 + 1/2^4) * (1 - 1/2^5) * ... = 0.568698946265428505954976737...

Cf. A000593, A027637, A048651, A062510, A065446, A079555, A081845, A085011, A100221, A132020, A330862, A216206.

RealDigits[QPochhammer[-1, -1/2]/2, 10, 110] [[1]]
N[3/QPochhammer[-2, 1/4], 120] (* Vaclav Kotesovec, Apr 28 2020 *)

prodinf(k=1, 1 + 1/(-2)^k) \\ Michel Marcus, Apr 28 2020

Equals Product_{k>=1} 1/(1 + 1/2^(2*k-1)).
Equals exp(Sum_{k>=1} A000593(k)/(k*(-2)^k)).
From Peter Bala, Dec 15 2020: (Start)
Constant C = (2/3) - (1/3)*Sum_{n >= 0} (-1)^n * 2^(n^2)/( Product_{k = 1..n+1} 4^k - 1 ).
C = Sum_{n >= 0} 1/( Product_{k = 1..n} (-2)^k - 1 ) = 1 - 1/3  - 1/9 + 1/81 + 1/1215 - - + + ... = Sum_{n >= 0} 1/A216206(n).
C = 1 + Sum_{n >= 0} (-1/2)^(n+1)*Product_{k = 1..n} (1 + (-1/2)^k).
3*C = 2 - Sum_{n >= 0} (1/4)^(n+1)*Product_{k = 1..n} (1 + (-1/2)^k).
9*C = 5 - Sum_{n >= 0} (-1/8)^(n+1)*Product_{k = 1..n} (1 + (-1/2)^k).
81*C = 46 + Sum_{n >= 0} (1/16)^(n+1)*Product_{k = 1..n} (1 + (-1/2)^k).
1215*C = 691 + Sum_{n >= 0} (-1/32)^(n+1)*Product_{k = 1..n} (1 + (-1/2)^k).
The sequence [1, 2, 5, 46, 691, ...] is the sequence of numerators of the  partial sums of the series Sum_{n >= 0} 1/A216206(n). (End)

A174608 Decimal expansion of (1/log(2))*Sum_{k>=0} log(1+1/2^k) = 2.253...

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Mar 23 2010

Brigitte Vallée, Digits and continuants in Euclidean algorithms. Ergodic vs tauberian theorems, Journal de théorie des nombres de Bordeaux 12 (2000), 519-558.

Cf. A002162, A081845.

digits = 105; 1/Log[2]*NSum[Log[1 + 1/2^k], {k, 0, Infinity}, WorkingPrecision -> digits, NSumTerms -> 70] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 15 2013 *)

Equals log(A081845)/A002162. - R. J. Mathar, Apr 20 2010

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A345987 Decimal expansion of constant mu(ell) arising in study of complexity of Euclidean algorithm.

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A371748 Decimal expansion of Product_{k>=0} (1 + 1/4^k).

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A371751 Decimal expansion of Product_{k>=0} (1 + 1/5^k).

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A005477 a(n) = 2^(n-1)*(2^n - 1)*Product_{j=1..n-1} (2^j + 1).

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A330863 Decimal expansion of Product_{k>=1} (1 + 1/(-2)^k).

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A174608 Decimal expansion of (1/log(2))*Sum_{k>=0} log(1+1/2^k) = 2.253...

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A005477 a(n) = 2^(n-1)(2^n - 1)Product_{j=1..n-1} (2^j + 1).