A340533 -id:A340533 - cp's OEIS frontend

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

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.

A341641 Decimal expansion of the probability of two consecutive continued fraction coefficients being both even, when the continued fraction coefficients satisfy the Gauss-Kuzmin distribution.

Original entry on oeis.org

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

Offset: 0

A.H.M. Smeets, Feb 16 2021

			0.1169400035780680765605607509208534105...

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

Cf. A340533, A340543.

sumpos(j=1, log(gamma(1+1/(4*j+2))/gamma(1+(j+1)/(2*j+1))*gamma(1+(2*j+1)/4/j)/gamma(1+1/4/j)))/log(2)

C = log(2)-1+(log(72*Pi)-4*log(gamma(1/4)))/log(2)
C+sumpos(n=2, (-1)^n*(zeta(n)-1)/n*((2^(2-n)-2^(2-2*n)-1)*(zeta(n)-1)+(2^(n-1)-1)*2^(2-2*n)))/log(2)

Equals Sum_{j >= 1} log_2(Gamma(1+1/(4*j+2))/Gamma(1+(j+1)/(2*j+1))*Gamma(1+(2*j+1)/4/j)/Gamma(1+1/4/j)).

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

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A341641 Decimal expansion of the probability of two consecutive continued fraction coefficients being both even, when the continued fraction coefficients satisfy the Gauss-Kuzmin distribution.

Views

Author

Keywords

Examples

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PARI

PARI

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