A019609 -id:A019609 - cp's OEIS frontend

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A159822 Continued fraction for Pi*e A019609.

8, 1, 1, 5, 1, 3, 1, 4, 12, 3, 2, 1, 5, 2, 12, 1, 1, 1, 10, 2, 2, 2, 3, 8, 3, 2, 2, 2, 29, 1, 1, 13, 1, 1, 8, 11, 16, 3, 1, 4, 163, 2, 1, 1, 1, 5, 1, 6, 1, 17, 5, 1, 3, 6, 3, 1, 4, 1, 1, 1, 5, 1, 7, 15, 4, 1, 1, 1, 9, 1, 1, 4, 1, 1, 9, 1, 55, 14, 14, 1, 3, 2, 3, 7, 1, 118, 1, 2, 29, 1, 2, 2, 1, 4, 1, 2, 1

Offset: 0

Harry J. Smith, Apr 27 2009

			Pi*e = 8.53973422267356706546... = 8 + 1/(1 + 1/(1 + 1/(5 + 1/(1 + ...))))

Harry J. Smith, Table of n, a(n) for n = 0..20000

ContinuedFraction[E*Pi,5! ] (* Vladimir Joseph Stephan Orlovsky, Jun 18 2009 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(Pi*exp(1)); for (n=1, 20001, write("b159822.txt", n-1, " ", x[n])); }

A019645 Decimal expansion of sqrt(Pi*e).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			2.9222823653222778645416230761076823153979...

Ivan Panchenko, Table of n, a(n) for n = 1..1000

Cf. A019609.

C := ComplexField(); [Sqrt(Pi(C)*Exp(1))]; // G. C. Greubel, Nov 17 2017

Mathematica

RealDigits[Sqrt[Pi E],10,120][[1]] (* Harvey P. Dale, Jun 14 2014 *)

PARI

sqrt(Pi*exp(1)) \\ G. C. Greubel, Nov 17 2017

Formula

Equals A002161*A019774. - R. J. Mathar, Apr 11 2024

A073244 Decimal expansion of Pi - e.

Original entry on oeis.org

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

Offset: 0

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Author

Rick L. Shepherd, Jul 21 2002

Keywords

cons
nonn

Examples

0.42331082513074800310235591192...

Crossrefs

Cf. A059742 (Pi+e), A000796 (Pi), A001113 (e), A019609 (Pi*e), A061382 (Pi/e), A061360 (e/Pi), A039661 (e^Pi), A059850 (Pi^e), A073233 (Pi^Pi), A073226 (e^e), A049006 (i^i = e^(-Pi/2)).

Cf. A110564 for continued fraction for Pi - e.

Programs

Mathematica

RealDigits[N[Pi-E,6! ]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jun 18 2009 *)

PARI

Pi-exp(1)

A131563 Decimal expansion of e*Pi*phi, where phi = (sqrt(5) + 1)/2.

Original entry on oeis.org

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

Offset: 2

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Author

Omar E. Pol, Aug 27 2007, Dec 17 2008

Keywords

cons
nonn

Examples

e*Pi*phi = 13.817580227...

Links

Harry J. Smith, Table of n, a(n) for n = 2..20000

Crossrefs

Decimal expansion of e: A001113. Decimal expansion of Pi: A000796. Decimal expansion of phi: A001622. e*Pi: A019609. Pi*phi: A094886. e*phi: A094885.

Programs

Maple

exp(1)*Pi*(1+sqrt(5))/2;

Mathematica

phi=(5^(1/2)+1)/2;RealDigits[N[Pi*E*phi,6! ]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jun 18 2009 *) RealDigits[E*Pi*GoldenRatio,10,120][[1]] (* Harvey P. Dale, Nov 02 2020 *)

PARI

{ default(realprecision, 20080); phi = (1 + sqrt(5))/2; x=exp(1)*Pi*phi/10; for (n=2, 20000, d=floor(x); x=(x-d)*10; write("b131563.txt", n, " ", d)); } \\ Harry J. Smith, Apr 26 2009

Extensions

More terms from N. J. A. Sloane, Dec 19 2008

A019610 Decimal expansion of Pi*e/2.

Original entry on oeis.org

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

Offset: 1

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Author

N. J. A. Sloane

Keywords

nonn
cons

Examples

4.26986711133678353273177543477328724751744426788255748...

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15 Glaisher-Kinkelin constant, p. 136.

Links

Ivan Panchenko, Table of n, a(n) for n = 1..1000

Z. A. Melzak, Infinite products for πe and π/e, Amer. Math. Monthly 68 (1961) 39-41.

Crossrefs

Cf. A019609, A241420, A241421.

Programs

Magma

SetDefaultRealField(RealField(100)); R:= RealField(); Pi(R)*Exp(1)/2; // G. C. Greubel, Aug 24 2018

Maple

Digits:=100: evalf(Pi*exp(1)/2); # Wesley Ivan Hurt, Aug 09 2014

Mathematica

RealDigits[(Pi*E)/2,10,120][[1]] (* Harvey P. Dale, Apr 16 2014 *)

PARI

{ default(realprecision, 100); x=(1/2)*Pi*exp(1); for(n=1, 100, d=floor(x); x=(x-d)*10; print1(d, ", ")) } \\ Altug Alkan, Nov 13 2015

Formula

Melzak's formula: lim_{n->infinity} Product_{k=1..2n+1} (1+2/k)^(k*(-1)^(k+1)) = Pi*e/2. - Jean-François Alcover, Apr 25 2014

A092036 Decimal expansion of (Pi*e)^2.

Original entry on oeis.org

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

Offset: 2

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Author

Mohammad K. Azarian, Mar 26 2004

Keywords

easy
nonn
cons

Examples

72.927060...

Crossrefs

Cf. A019609 (Pi*e).

Programs

Mathematica

RealDigits[N[(Pi*E)^2,6! ]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jun 18 2009 *)

PARI

Pi^2*exp(2) \\ Michel Marcus, Jan 17 2015

A131566 Decimal expansion of (e*Pi*phi)^2.

Original entry on oeis.org

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

Offset: 3

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Author

Omar E. Pol, Aug 27 2007

Keywords

cons
nonn

Comments

phi = (5^(1/2) + 1)/2 = (1 + sqrt(5))/2.

Examples

190.925523334...

Links

Harry J. Smith, Table of n, a(n) for n = 3..20000

Crossrefs

Cf. A001113, A000796, A001622, A019609, A094886, A094885.

Programs

Mathematica

phi=(5^(1/2)+1)/2;RealDigits[N[(Pi*E*phi)^2,6! ]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jun 18 2009 *) RealDigits[(E*Pi*GoldenRatio)^2,10,120][[1]] (* Harvey P. Dale, May 01 2017 *)

PARI

{ default(realprecision, 20080); phi = (1 + sqrt(5))/2; x=(exp(1)*Pi*phi)^2/100; for (n=3, 20000, d=floor(x); x=(x-d)*10; write("b131566.txt", n, " ", d)); } \\ Harry J. Smith, Apr 27 2009

Extensions

More terms from Harry J. Smith, Apr 26 2009

Fixed my PARI program, had -n Harry J. Smith, May 19 2009

A121246 a(n) = ceiling((Pi^n)*(e^n)).

Original entry on oeis.org

9, 73, 623, 5319, 45418, 387853, 3312154, 28284913, 241545634, 2062735515, 17615213066, 150429237854, 1284625710592, 10970362144063, 93683977036771, 800036264817078, 6832097070038331, 58344293161634134, 498244757010106386

Offset: 1

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Author

Mohammad K. Azarian, Aug 22 2006

Keywords

nonn
easy
less

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A019609, A062541.

Programs

Maple

A121246:=n->ceil((Pi^n)*(exp(1)^n)): seq(A121246(n), n=1..30); # Wesley Ivan Hurt, Jan 30 2017

Mathematica

Table[Ceiling[Pi^n E^n],{n,20}] (* Harvey P. Dale, Aug 20 2012 *)

A203816 Decimal expansion of e*gamma*Pi, where gamma is Euler's constant.

Original entry on oeis.org

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

Offset: 1

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Author

Yalcin Aktar, Jan 06 2012

Keywords

nonn
cons

Examples

4.929268367422897891526304798034231...

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001620 (Euler's constant), A019609 (e*Pi).

Programs

Magma

SetDefaultRealField(RealField(100)); R:= RealField(); Exp(1)*Pi(R)*EulerGamma(R); // G. C. Greubel, Sep 03 2018

Mathematica

RealDigits[E*EulerGamma*Pi, 10, 105][[1]] (* Alonso del Arte, Jan 06 2012 *)

PARI

exp(1)*Euler*Pi \\ Charles R Greathouse IV, Jan 15 2012

A096411 Decimal expansion of 1/sqrt(Pi*e).

Original entry on oeis.org

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

Offset: 0

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Author

Mohammad K. Azarian, Aug 07 2004

Keywords

cons
nonn

Examples

0.3421982803122165331792511834700...

Links

G. C. Greubel, Table of n, a(n) for n = 0..5000

Crossrefs

Cf. A000796, A001113, A019609.

Programs

Mathematica

RealDigits[N[1/(Pi*E)^(1/2),6! ]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jun 18 2009 *)

PARI

1/sqrt(Pi*exp(1)) \\ G. C. Greubel, Jan 11 2017

Extensions

Definition clarified by R. J. Mathar, Feb 05 2009

Showing 1-10 of 41 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.
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cp's OEIS Frontend

A159822 Continued fraction for Pi*e A019609.

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Author

Keywords

Examples

Links

Programs

Mathematica

PARI

A019645 Decimal expansion of sqrt(Pi*e).

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Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A073244 Decimal expansion of Pi - e.

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Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

A131563 Decimal expansion of e*Pi*phi, where phi = (sqrt(5) + 1)/2.

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Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A019610 Decimal expansion of Pi*e/2.

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Author

Keywords

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A092036 Decimal expansion of (Pi*e)^2.

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Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

A131566 Decimal expansion of (e*Pi*phi)^2.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A121246 a(n) = ceiling((Pi^n)*(e^n)).

Views

A131563 Decimal expansion of ePiphi, where phi = (sqrt(5) + 1)/2.

A131566 Decimal expansion of (ePiphi)^2.

A203816 Decimal expansion of egammaPi, where gamma is Euler's constant.