A113014 -id:A113014 - cp's OEIS frontend

A365105 Continued fraction expansion of 1/(2+3/(4+5/(6+7/(...)))) = A113014.

0, 2, 1, 1, 1, 2, 1, 2, 10, 2, 2, 66, 1, 1, 13, 66, 9, 5, 8, 1, 9, 1, 1, 1, 5, 2, 2, 3, 1, 1, 1, 16, 99, 6, 1, 5, 1, 2, 1, 55, 2, 2, 1, 1, 6, 4, 1, 1, 1, 40, 1, 1, 1, 6, 14, 7, 9, 1, 1, 2, 3, 2, 2, 2, 1, 1, 2, 7, 12, 1, 2, 2, 1, 4, 2, 4, 2, 1, 3, 2, 1, 10, 7, 1, 4, 1, 119, 1, 1, 1, 3, 5, 2, 12, 1

Offset: 0

Rok Cestnik, Aug 21 2023

A113014 is defined by a generalized continued fraction and the expansion here is its simple continued fraction.

			1/(2+1/(1+1/(1+1/(1+1/(2+1/(...)))))) = 1/(2+3/(4+5/(6+7/(...)))).

Cf. A113014, A365116.

A365105 = ContinuedFraction[Sqrt[2*E/Pi]/Erfi[1/Sqrt[2]]-1,#]&;

A365116 Greedy Egyptian fraction expansion of 1/(2+3/(4+5/(6+7/(...)))) = A113014.

Original entry on oeis.org

3, 22, 1060, 1471180, 4470565318951, 21387196871513452925199541, 508406155285302398938678134812723800438323137635884, 293206664431782985993415302840424364324000216460330294129310314248712028637333711113413230006858255456

Offset: 1

Rok Cestnik, Aug 22 2023

nonn

			1/3 + 1/22 + 1/1060 + 1/1471180 + ... = 1/(2+3/(4+5/(6+7/(...)))).

Cf. A113014, A365105.

A306858 Decimal expansion of 1 - 1/(13) + 1/(135) - 1/(1357) + ...

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Jun 24 2019

			0.7247784590070763318182279676062161663121329...

Cf. A001147, A014481, A060196, A068996, A092605, A113014.

RealDigits[Sqrt[Pi/(2 Exp[1])] Erfi[1/Sqrt[2]], 10, 110] [[1]]
RealDigits[Sqrt[2] DawsonF[1/Sqrt[2]], 10, 110] [[1]]

Equals sqrt(Pi/(2*exp(1)))*erfi(1/sqrt(2)), where erfi is the imaginary error function.
Equals (1/sqrt(e)) * Sum_{k>=0} 1/(2^k * k! * (2*k+1)) = 1/(sqrt(e)) * Sum_{k>=0} 1/A014481(k). - Amiram Eldar, Nov 12 2021
Equals 1/(1+A113014). - Jon Maiga, Nov 12 2021

A367120 Decimal expansion of continued fraction 2+1/(4+3/(6+5/(8+7/(...)))).

Original entry on oeis.org

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

Offset: 1

Rok Cestnik, Nov 13 2023

			2.224412437956340467163837541384021939...

Cf. A113014, A113011, A194807, A365307.

First[RealDigits[2/HypergeometricPFQ[{1, 1}, {3/2, 3}, -1/2], 10, 100]] (* or *)
First[RealDigits[2/Sum[(-1)^k/Binomial[k+2, 2]/(2*k+1)!!, {k, 0, Infinity}], 10, 100]] (* Paolo Xausa, Nov 18 2024 *)

N=50;
doblfac(n) = if(n<0, 0, n<2, 1, n*doblfac(n-2));
ap1 = 2 / sum(k=0,N, (-1)^k/binomial(k+2,2)/doblfac(2*k+1));
ap2 = 2 / sum(k=0,N+1, (-1)^k/binomial(k+2,2)/doblfac(2*k+1));
n = 0; while(digits(floor(10^(n+1)*ap1)) == digits(floor(10^(n+1)*ap2)), n++);
A367120 = digits(floor(10^n*ap1));

Equals 2 / pFq(1,1; 3/2,3; -1/2) where pFq() is the generalized hypergeometric function.

Equals 2 / Sum_{k>=0} (-1)^k/binomial(k+2,2)/(2*k+1)!! = 2 / (1 - 1/9 + 1/90 - 1/1050 + 1/14175 - 1/218295 + ... ).

cp's OEIS Frontend

A365105 Continued fraction expansion of 1/(2+3/(4+5/(6+7/(...)))) = A113014.

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A365116 Greedy Egyptian fraction expansion of 1/(2+3/(4+5/(6+7/(...)))) = A113014.

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A306858 Decimal expansion of 1 - 1/(13) + 1/(135) - 1/(1357) + ...

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Formula

A367120 Decimal expansion of continued fraction 2+1/(4+3/(6+5/(8+7/(...)))).

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cp's OEIS Frontend

A365105 Continued fraction expansion of 1/(2+3/(4+5/(6+7/(...)))) = A113014.

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Mathematica

A365116 Greedy Egyptian fraction expansion of 1/(2+3/(4+5/(6+7/(...)))) = A113014.

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Author

Keywords

Examples

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A306858 Decimal expansion of 1 - 1/(1*3) + 1/(1*3*5) - 1/(1*3*5*7) + ...

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A367120 Decimal expansion of continued fraction 2+1/(4+3/(6+5/(8+7/(...)))).

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Author

Keywords

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Mathematica

PARI

Formula

A306858 Decimal expansion of 1 - 1/(13) + 1/(135) - 1/(1357) + ...