cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A083864 Decimal expansion of Product_{k>=0} (1 - 1/(2^k+1)).

Original entry on oeis.org

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

Views

Author

Benoit Cloitre, Jun 19 2003

Keywords

Comments

c/4 where c is the constant defined in A085011.

Examples

			0.2097112208975537988549780538514871...

Crossrefs

Cf. A081845, A085011, A261584.

Programs

  • Mathematica
    RealDigits[1/QPochhammer[-1, 1/2], 10, 120][[1]] (* Amiram Eldar, May 29 2023 *)
  • PARI
    prod(k=0,1000,1-1./(2^k+1))
  • PARI
    prodinf(k=0, 1-1/(2^k+1)) \\ Michel Marcus, Feb 28 2020

Formula

Product_{k>=0} (1-1/(2^k+1)).
From Robert FERREOL, Feb 28 2020: (Start)
Equals Product_{k>=0} (1 + 1/2^k)^(-1) = 1/A081845.
Equals 1 + Sum_{k>=1} (-1)^k*2^(k*(k+1)/2)/((2-1)*(2^2-1)*...*(2^k-1)). (End)
From Peter Bala, Jan 16 2021: (Start)
Constant C = 2^(-1)*Sum_{n >= 0} (-1/2)^n/Product_{k = 1..n} (1 - 1/2^k).
C = (2^2/(3*5))*Sum_{n >= 0} (-1/8)^n/Product_{k = 1..n} (1 - 1/2^k).
C = (2^9/(3*5*9*17))*Sum_{n >= 0} (-1/32)^n/Product_{k = 1..n} (1 - 1/2^k).
C = (2^20/(3*5*9*17*33*65))*Sum_{n >= 0} (-1/128)^n/Product_{k = 1..n} (1 - 1/2^k) and so on. (End)

A330862 Decimal expansion of Product_{k>=1} (1 - 1/(-2)^k).

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Apr 28 2020

Keywords

Examples

			(1 + 1/2) * (1 - 1/2^2) * (1 + 1/2^3) * (1 - 1/2^4) * (1 + 1/2^5) * ... = 1.2107241303010591801361728561...

Crossrefs

Cf. A000203, A048651, A065446, A079555, A081845, A085011, A100221, A132020, A330863.

Programs

  • Mathematica
    RealDigits[QPochhammer[-1/2, -1/2], 10, 111] [[1]]
N[QPochhammer[-2, 1/4]*QPochhammer[1/4]/3, 120] (* Vaclav Kotesovec, Apr 28 2020 *)
  • PARI
    prodinf(k=1, 1 - 1/(-2)^k) \\ Michel Marcus, Apr 28 2020

Formula

Equals Product_{k>=1} (4^k - 1)*(4^k + 2)/4^(2*k).
Equals exp(-Sum_{k>=1} A000203(k)/(k*(-2)^k)).

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

Views

Author

Ilya Gutkovskiy, Apr 28 2020

Keywords

Examples

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

Crossrefs

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

Programs

  • Mathematica
    RealDigits[QPochhammer[-1, -1/2]/2, 10, 110] [[1]]
N[3/QPochhammer[-2, 1/4], 120] (* Vaclav Kotesovec, Apr 28 2020 *)
  • PARI
    prodinf(k=1, 1 + 1/(-2)^k) \\ Michel Marcus, Apr 28 2020

Formula

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)

A084118 Continued fraction expansion of Product_{k>=0} (1 - 1/(2^k+1)).

Original entry on oeis.org

0, 4, 1, 3, 3, 7, 2, 1, 1, 3, 3, 18, 9, 15, 1, 13, 3, 1, 1, 1, 4, 3, 1, 3, 1, 1, 1, 1, 3, 8, 18, 2, 3, 2, 14, 3, 1, 2, 1, 7, 1, 300, 1, 3, 1, 17, 1, 1, 1, 4, 3, 9, 2, 3, 4, 1, 25, 6, 12, 3, 1, 1, 3, 4, 2, 6, 28, 2, 11, 2, 3, 1, 2, 3, 18, 9, 1, 1, 3, 2, 22, 165, 8, 5, 13, 7, 1, 6, 1, 2, 3, 5, 1, 14, 6, 1
Offset: 0

Views

Author

Benoit Cloitre, Jun 19 2003

Keywords

Comments

c/4 where c is the constant defined in A085011.

Crossrefs

Cf. A085011.

Programs

  • PARI
    contfrac(prod(k=0,1000,1-1./(2^k+1)))

Formula

Product_{k>=0} (1 - 1/(2^k+1)) = 0.2097112208975537988549780538514871...

A084129 Continued fraction expansion of 4*Product_{k>=0} (1 - 1/(2^k+1)).

Original entry on oeis.org

0, 1, 5, 4, 1, 6, 1, 8, 13, 4, 1, 1, 8, 1, 1, 3, 2, 13, 1, 3, 2, 3, 4, 2, 3, 2, 14, 2, 73, 1, 2, 1, 58, 14, 1, 1, 4, 75, 5, 73, 1, 1, 2, 1, 2, 1, 1, 1, 1, 6, 20, 1, 5, 1, 1, 5, 1, 1, 2, 1, 1, 3, 14, 1, 8, 1, 1, 1, 27, 1, 46, 1, 3, 5, 73, 2, 2, 1, 1, 3, 1, 2, 5, 3, 1, 40, 1, 1, 7, 1, 2, 3, 3, 31, 2, 12, 3, 1
Offset: 0

Views

Author

Benoit Cloitre, Jun 19 2003

Keywords

Crossrefs

Cf. A085011.

Programs

  • PARI
    contfrac(4*prod(k=0,1000,1-1./(2^k+1)))

Formula

4*Product_{k>=0} (1 - 1/(2^k+1)) = 0.838844883590215195...
Showing 1-5 of 5 results.

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