A063448 -id:A063448 - cp's OEIS frontend

A063447 Continued fraction for Pi * sqrt(2).

4, 2, 3, 1, 7, 7, 1, 3, 1, 1, 1, 1, 4, 10, 8, 1, 2, 3, 3, 2, 5, 8, 6, 14, 1, 9, 1, 1, 1, 2, 6, 2, 2, 4, 3, 2, 2, 6, 1, 12, 1, 35, 32, 1, 3, 5, 15, 1, 2, 1, 6, 1, 2, 1, 1, 2, 16, 6, 1, 7, 1, 2, 2, 1, 2, 1, 1, 27, 3, 6, 4, 26, 2, 1, 31, 2, 1, 1, 12, 1, 1, 2, 2, 1, 24, 5, 2, 591, 6, 33, 1, 8, 1, 2, 6, 2

Offset: 0

Jason Earls, Jul 24 2001

			4.442882938158366247015880990... = 4 + 1/(2 + 1/(3 + 1/(1 + 1/(7 + ...))))

Harry J. Smith, Table of n, a(n) for n = 0..19999

Cf. A063448 (decimal expansion).

SetDefaultRealField(RealField(150)); R:= RealField(); ContinuedFraction(Pi(R)*Sqrt(2)); // G. C. Greubel, Aug 16 2018

ContinuedFraction[Pi*Sqrt[2],300] (* Vladimir Joseph Stephan Orlovsky, Mar 21 2011*)

PARI
```
contfrac(Pi*sqrt(2))
```

allocatemem(932245000); default(realprecision, 21000); x=contfrac(Pi*sqrt(2)); for (n=1, 20000, write("b063447.txt", n-1, " ", x[n])) \\ Harry J. Smith, Aug 21 2009

Offset changed by Andrew Howroyd, Aug 04 2024

A343392 Decimal expansion of 2Pisqrt(2).

Original entry on oeis.org

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

Offset: 1

Bernard Schott, Apr 13 2021

Circumference of the circumcircle of the square whose sides = 2.
Hypotenuse of the right isosceles triangle with the two legs = 2*Pi.
Perimeter of the closed curve with implicit Cartesian equation x^2 + y^2 = abs(x) + abs(y). This curve in the first quadrant is the half-circle with equation (x-1/2)^2 + (y-1/2)^2 = 1/2, hence, the curve is the union of 4 identical half-circles with diameter = sqrt(2) obtained by symmetries. (See link Curve.)
S. Ramanujan produced a curious approximation to 2*Pi*sqrt(2) by dividing 99^2 by 1103 (see link Prime Curios! and A343393).

			8.88576587631673249403176198012138739722924337875138044617

David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 54.

Chris K. Caldwell and G. L. Honaker, Jr., 1103, 1st comment, Prime Curios!
Bernard Schott, Curve x^2+y^2 = abs(x)+abs(y).
Index to sequences related to curves.
Index entries for transcendental numbers.

Cf. A000796, A002193, A010466, A019692, A063448, A343393.

Maple
```
evalf(2*Pi*sqrt(2),120);
```

RealDigits[2*Sqrt[2]*Pi, 10, 100][[1]] (* Amiram Eldar, Apr 13 2021 *)

2*Pi*sqrt(2) = A019692 * A002193 = A010466 * A000796 = 2 * A063448.

A217458 Decimal expansion of 2^(Pi*sqrt(2)).

Original entry on oeis.org

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

Offset: 2

Jani Melik, Oct 03 2012

			21.7490870543774583340992082660113952338588408897418426759...

G. C. Greubel, Table of n, a(n) for n = 2..10000

Cf. A063448 (Pi * sqrt(2)).

SetDefaultRealField(RealField(100)); R:=RealField(); 2^(Pi(R)*Sqrt(2)); // G. C. Greubel, Oct 05 2018

RealDigits[2^(Pi*Sqrt[2]), 10, 100][[1]] (* G. C. Greubel, Oct 05 2018 *)

fpprec : 100; ev(bfloat(2^(%pi*sqrt(2)))); /* Martin Ettl, Oct 04 2012 */

default(realprecision, 100); 2^(Pi*sqrt(2)) \\ G. C. Greubel, Oct 05 2018

Sage
```
2^(pi*sqrt(2)).n(digits=100)
```

A359533 Decimal expansion of Sum_{k>=0} (-1/64)^kbinomial(2k, k)^3(4k + 1)*H_k, where H_k is the k-th harmonic number (negated).

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Jan 04 2023

			0.276427204245986573092639825616889946783740795...

John M. Campbell, Applications of a class of transformations of complex sequences, arXiv:2212.13305 [math.NT], 2022. See p. 6.

Cf. A000796, A001008, A002805, A002897, A016639, A016813, A063448, A068465, A068466, A091925, A203142, A203143.

Cf. A359532.

First[RealDigits[N[(Gamma[1/8]Gamma[3/8]/(Gamma[1/4]Gamma[3/4]))^2/(6Sqrt[2]Pi)-4Log[2]/Pi,100]]]

Equals 4*log(2)/Pi - (Gamma(1/8)*Gamma(3/8)/(Gamma(1/4)*Gamma(3/4)))^2/(6*sqrt(2)*Pi).

Equals 4*log(2)/Pi - (Gamma(1/8)*Gamma(3/8))^2/(12*sqrt(2)*Pi^3).

A376207 Numbers k such that ceiling(2Pik/sqrt(2)) != ceiling(Pi/arcsin(sqrt(2)/(2*k))).

Original entry on oeis.org

1, 70, 569, 58704, 15770314

Offset: 1

Hugo Pfoertner, Sep 15 2024

2*n/sqrt(2) > 1/arcsin(sqrt(2)/(2*n)) for all n > 0.

Limit_{x->oo} 2*x/sqrt(2) - 1/arcsin(sqrt(2)/(2*x)) = 0.

			  n    k=a(n)        2*Pi*k/sqrt(2)   Pi/arcsin(sqrt(2)/(2*k))
  1         1         4.44288293816             4.000000000000
  2        70       311.00180567109           310.996516371805
  3       569      2528.00039181211          2527.999741125982
  4     58704    260815.00000164873        260814.999995341832
  5  15770314  70065659.00000001744      70065658.999999993965

Cf. A063448, A376066.

Cf. A120702.

A347055 Decimal expansion of Pi * (sqrt(3) - sqrt(2)).

Original entry on oeis.org

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

Offset: 0

Sean A. Irvine, Aug 14 2021

			0.9985151545442873047663537828657734982375770549025315...

Michael I. Shamos, Shamos's catalog of the real numbers (2011).
Index entries for transcendental numbers.

Cf. A063448, A304656.

RealDigits[Pi * (Sqrt[3] - Sqrt[2]), 10, 120][[1]] (* Amiram Eldar, Jun 20 2023 *)

Pi*(sqrt(3)-sqrt(2)) \\ Charles R Greathouse IV, Oct 02 2022

Equals Integral_{x>=0} log(1 + 1/(x^2+2)) dx.

cp's OEIS Frontend

A063447 Continued fraction for Pi * sqrt(2).

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A343392 Decimal expansion of 2*Pi*sqrt(2).

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Maple

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A217458 Decimal expansion of 2^(Pi*sqrt(2)).

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Maxima

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A359533 Decimal expansion of Sum_{k>=0} (-1/64)^k*binomial(2*k, k)^3*(4*k + 1)*H_k, where H_k is the k-th harmonic number (negated).

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A376207 Numbers k such that ceiling(2*Pi*k/sqrt(2)) != ceiling(Pi/arcsin(sqrt(2)/(2*k))).

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A347055 Decimal expansion of Pi * (sqrt(3) - sqrt(2)).

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Author

Keywords

Examples

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Crossrefs

Programs

Mathematica

PARI

Formula

A343392 Decimal expansion of 2Pisqrt(2).

A359533 Decimal expansion of Sum_{k>=0} (-1/64)^kbinomial(2k, k)^3(4k + 1)*H_k, where H_k is the k-th harmonic number (negated).

A376207 Numbers k such that ceiling(2Pik/sqrt(2)) != ceiling(Pi/arcsin(sqrt(2)/(2*k))).