A216582 -id:A216582 - cp's OEIS frontend

A340533 Decimal expansion of log_2(4/Pi).

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

Offset: 0

A.H.M. Smeets, Jan 10 2021

Probability of a coefficient in the continued fraction being even, where the continued fraction coefficients satisfy the Gauss-Kuzmin distribution.

			0.348503870527681201956720704891992664981523...

V. N. Nolte, Some probabilistic results on the convergents of continued fractions, Indagationes Mathematicae, Vol. 1, No. 3 (1990), pp. 381-389.

Cf. A088538, A216582, A340543.

RealDigits[Log[4/Pi]/Log[2], 10, 100][[1]] (* Amiram Eldar, Jan 10 2021 *)

PARI
```
log(4/Pi)/log(2)
```

Equals 2 - A216582.
Equals log_2(A088538).
Equals -Sum_{k >= 1} log_2(1-1/(2*k+1)^2).
Equals 1-A340543.

A340543 Decimal expansion of log(Pi/2)/log(2).

Original entry on oeis.org

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

Offset: 0

A.H.M. Smeets, Jan 11 2021

Probability of a coefficient in the continued fraction being odd, where the continued fraction coefficients satisfy the Gauss-Kuzmin distribution.

			0.65149612947231879804327929510800733501847692676304...

V. N. Nolte, Some probabilistic results on the convergents of continued fractions, Indagationes Mathematicae, Vol. 1, No. 3 (1990), pp. 381-389.

Cf. A019669. Essentially the same as A216582.

Cf. A340533.

RealDigits[Log2[Pi/2], 10, 120][[1]] (* Amiram Eldar, May 29 2023 *)

PARI
```
log(Pi/2)/log(2)
```

Equals A216582 - 1.
Equals log_2(A019669).
Equals Sum_{k >= 0} -log_2(1-1/(2*k+2)^2).
Equals 1-A340533.

cp's OEIS Frontend

A340533 Decimal expansion of log_2(4/Pi).

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