A039661 -id:A039661 - cp's OEIS frontend

A063503 Continued fraction for e^Pi - Pi^e (A063504 = A039661 - A059850).

0, 1, 2, 7, 7, 6, 2, 1, 6, 2, 5, 7, 1, 3, 1, 1, 6, 1, 1, 1, 1, 1, 2, 1, 11, 5, 6, 2, 1, 124, 1, 4, 2, 1, 1, 3, 18, 1, 1, 1, 1, 17, 1, 2, 10, 1, 1, 1, 2, 2, 2, 2, 3, 1, 2, 4, 84, 1, 1, 1, 4, 1, 1, 15, 2, 1, 1, 17, 1, 1, 8, 1, 1, 10, 1, 3, 1, 2, 2, 1, 2, 1, 2, 4, 22, 4, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 9

Offset: 0

Robert G. Wilson v, Jul 30 2001

			0.6815349144182235323019341634048123526710...

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

Cf. A063504 (decimal expansion).

Mathematica
```
ContinuedFraction[E^Pi - Pi^E, 100]
```

{ allocatemem(932245000); default(realprecision, 21000); e=exp(1); x=contfrac(e^Pi - Pi^e); for (n=1, 20000, write("b063503.txt", n-1, " ", x[n])) } \\ Harry J. Smith, Aug 24 2009

Offset changed by Andrew Howroyd, Aug 04 2024

A042972 Decimal expansion of i^(-i), where i = sqrt(-1).

Original entry on oeis.org

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

Offset: 1

Marvin Ray Burns

Square root of Gelfond's constant (A039661). Since Gelfond's constant e^Pi is transcendental, e^(Pi/2) is transcendental. - Daniel Forgues, Apr 15 2011
The complex sequence (...((((i)^i)^i)^i)^...) (n pairs of brackets) is periodic with period 4 and the first four entries are i, e^(-Pi/2), -i, e^(+Pi/2). See A049006 for e^(-Pi/2). - Wolfdieter Lang, Apr 27 2013
A solution of x^i + x^(-i) = 0. In fact, x = Exp(Pi/2 + k*Pi), where k is any integer. - Robert G. Wilson v, Feb 04 2014

			4.81047738096535165547303566670383312639017087466453494002...

Nathaniel Johnston, Table of n, a(n) for n = 1..10000
H. S. Uhler, On the numerical value of i^i, Amer. Math. Monthly, 28 (1921), 114-116.
Index entries for transcendental numbers

Cf. A049006.

SetDefaultRealField(RealField(110)); R:= RealField(); Exp(Pi(R)/2); // G. C. Greubel, Feb 17 2019

evalf[110](I^(-I)); # Muniru A Asiru, Feb 17 2019

RealDigits[Re[I^(1/I)], 10, 100][[1]] (* Alonso del Arte, Oct 31 2011 *)
RealDigits[ Exp[Pi/2], 10, 111][[1]] (* Robert G. Wilson v, Apr 08 2014 *)

default(realprecision, 110); exp(Pi/2) \\ Nathaniel Johnston, Apr 15 2011 (modified by G. C. Greubel, Feb 17 2019)

numerical_approx(exp(pi/2), digits=110) # G. C. Greubel, Feb 17 2019

Equals i^(-i) = i^(1/i) = e^(Pi/2).

Also (((i)^i)^i)^i. See a comment above on such powers. - Wolfdieter Lang, Apr 27 2013

a(100) corrected by Nathaniel Johnston, Apr 15 2011

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

Rick L. Shepherd, Jul 21 2002

			0.42331082513074800310235591192...

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.

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

PARI
```
Pi-exp(1)
```

A093580 Decimal expansion of e^(-Pi).

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, May 14 2004

Also, decimal expansion of (-1)^i. - Rick L. Shepherd, Jul 09 2013

Also, the greatest real value of z that minimizes z^i + z^(-i). - Colin Linzer, Nov 21 2024

			0.04321391826377224977441773717172801127572810981...

Harry J. Smith, Table of n, a(n) for n = 0..20000
H. S. Uhler, On the numerical value of i^i, Amer. Math. Monthly, 28 (1921), 114-116.
Index entries for transcendental numbers.

Cf. A039661, A001113, A000796, A049006.

Join[{0}, RealDigits[E^-Pi, 10, 105][[1]]] (* Harvey P. Dale, Apr 25 2012 *)

{ default(realprecision, 20080); x=10*exp(-Pi); for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b093580.txt", n, " ", d)); } \\ Harry J. Smith, Jun 19 2009

Equals 1/A039661 = A049006^2. - Hugo Pfoertner, Nov 22 2024

A063504 Decimal expansion of e^Pi - Pi^e.

Original entry on oeis.org

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

Offset: 0

Robert G. Wilson v, Jul 30 2001

A classic calculus analysis problem is to discover whether e^Pi or Pi^e is the greater without the use of a calculator.

			0.681534914418223532301934163404812352676791108603519744242043855457416... - _Harry J. Smith_, Aug 24 2009

Paul J. Nahin, When Least Is Best, How Mathematicians Discovered Many Clever Ways to Make Things as Small (or as Large) as Possible, Princeton University Press, Princeton NJ, 2004, Page 144.
Alfred S. Posamentier & Ingmar Hehmann, Pi: A Biography of the World's Most Mysterious Number, Prometheus Books, NY 2002, pages 146, 301-304.

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

Equals A039661 - A059850.

Cf. A063503.

Mathematica
```
RealDigits[N[E^Pi - Pi^E, 100]][[1]]
```

{ default(realprecision, 20080); e=exp(1); x=10*(e^Pi - Pi^e); for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b063504.txt", n, " ", d)) } \\ Harry J. Smith, Aug 24 2009

Offset corrected by R. J. Mathar, Feb 05 2009

A058287 Continued fraction for e^Pi.

Original entry on oeis.org

23, 7, 9, 3, 1, 1, 591, 2, 9, 1, 2, 34, 1, 16, 1, 30, 1, 1, 4, 1, 2, 108, 2, 2, 1, 3, 1, 7, 1, 2, 2, 2, 1, 2, 3, 2, 166, 1, 2, 1, 4, 8, 10, 1, 1, 7, 1, 2, 3, 566, 1, 2, 3, 3, 1, 20, 1, 2, 19, 1, 3, 2, 1, 2, 13, 2, 2, 11, 3, 1, 2, 1, 7, 2, 1, 1, 1, 2, 1, 19, 1, 1, 12, 11, 1, 4, 1, 6, 1, 2, 18, 1, 2

Offset: 0

Robert G. Wilson v, Dec 07 2000

"The transcendentality of e^{Pi} was proved in 1929." (Wells)

			e^Pi = 23.140692632779269005... = 23 + 1/(7 + 1/(9 + 1/(3 + 1/(1 + ...)))). - _Harry J. Smith_, Apr 19 2009

Jan Gullberg, "Mathematics, From the Birth of Numbers," W. W. Norton and Company, NY and London, 1997, page 86.
David Wells, "The Penguin Dictionary of Curious and Interesting Numbers," Revised Edition, Penguin Books, London, England, 1997, page 81.

Harry J. Smith, Table of n, a(n) for n = 0..20000
V. Yu. Irkhin, Relations between e and Pi: Nilakantha's series and Stirling's formula, arXiv:2206.07174 [math.HO], 2022.
G. Xiao, Contfrac
Index entries for continued fractions for constants

Cf. A039661.

with(numtheory): cfrac(evalf((exp(1))^(evalf(Pi)),2560),256,'quotients');

Mathematica
```
ContinuedFraction[ E^Pi, 100]
```
PARI
```
\p 300 contfrac(exp(1)^Pi)
```

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(exp(1)^Pi); for (n=0, 20000, write("b058287.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Apr 19 2009

More terms from Jason Earls, Jun 21 2001

A175314 Decimal expansion of exp(Pi) + exp(-Pi).

Original entry on oeis.org

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

Offset: 2

R. J. Mathar, Apr 01 2010

			23.1839065510430412555035041051202753...

Index entries for transcendental numbers.

Cf. A039661 (exp(Pi)), A093580 (exp(-Pi)), A175315, A334402 (cosh(Pi)).

Maple
```
evalf(exp(Pi)+exp(-Pi)) ;
```

RealDigits[Exp[Pi]+Exp[-Pi],10,120][[1]] (* Harvey P. Dale, Jan 28 2017 *)

exp(Pi) + exp(-Pi) \\ Michel Marcus, Jun 27 2019

2*cosh(Pi) \\ Charles R Greathouse IV, Jan 30 2025

Equals A039661 + A093580.
Equals 2*cosh(Pi).
Equals 10 * Product_{k>=1} (1 + 4/(2*k+1)^2). - Amiram Eldar, Aug 09 2020

A166748 E.g.f.: exp(6*arcsin(x)).

Original entry on oeis.org

1, 6, 36, 222, 1440, 9990, 74880, 609390, 5391360, 51798150, 539136000, 6060383550, 73322496000, 951480217350, 13198049280000, 195053444556750, 3061947432960000, 50908949029311750, 894088650424320000

Offset: 0

Jaume Oliver Lafont, Oct 21 2009

nonn

exp(6*arcsin(1/2)) is Aleksandr Gelfond's constant exp(Pi).

G. C. Greubel, Table of n, a(n) for n = 0..445
A. R. Povolotsky et al., With regards to OEIS A166748, sci.math.symbolic usenet group, 2009

Cf. A166741, A006228, A039661.

Round[Table[3*2^(n-1)*(E^(3*Pi)-(-1)^n*E^(-3*Pi))*Abs[Gamma[n/2+3*I]]^2/Pi,{n,0,20}]] (* Vaclav Kotesovec, Nov 06 2014 *)
CoefficientList[Series[Exp[6*ArcSin[x]], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Nov 06 2014 *)

A166748(n)=round(norm(gamma(n/2+3*I))/Pi*if(n%2,cosh(3*Pi),sinh(3*Pi))*3<M. F. Hasler, Oct 25 2009

PARI
```
a(n)=polcoeff(exp(6*asin(x)),n)*n!
```

a(n)=(1+5*(n%2))*prod(k=0,n\2-1,(2*k+n%2)^2+36) \\ Jaume Oliver Lafont, Oct 28 2009

Contribution from Alexander R. Povolotsky, Oct 24 2009: (Start)
a(n+2) = (n^2+36)*a(n), a(0)=1, a(1)=6.
The above recurrence leads to
a(n) = (3*2^n*gamma(-3*i+n/2)*gamma(3*i+n/2)*(cos((n*Pi)/2)+i*sin((n*Pi)/2))*sinh(((6-i*n)*Pi)/2))/Pi where "i" is imaginary unit. (End)
a(n) = 3*2^(n-1)*(exp(3*Pi)-(-1)^n*exp(-3*Pi))*|Gamma(n/2+3i)|^2/Pi. - R. J. Mathar and M. F. Hasler, Oct 25 2009
a(n) ~ 6 * (exp(3*Pi) - (-1)^n*exp(-3*Pi)) * n^(n-1) / exp(n). - Vaclav Kotesovec, Nov 06 2014

Minor edits by Vaclav Kotesovec, Nov 06 2014

A175315 Decimal expansion of exp(Pi) - exp(-Pi).

Original entry on oeis.org

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

Offset: 2

R. J. Mathar, Apr 01 2010

			23.0974787145154967559546686307...

N. Kurokawa, M. Wakayama, A generalization of Lerch’s formula, Czechoslovak Math. J. 54 (2004), 941-947.
Index entries for transcendental numbers.

Cf. A039661 (exp(Pi)), A093580 (exp(-Pi)), A175314.

Maple
```
evalf(exp(Pi)-exp(-Pi)) ;
```

RealDigits[Exp[Pi]-Exp[-Pi],10,120][[1]] (* Harvey P. Dale, Aug 28 2019 *)

exp(Pi) + exp(-Pi) \\ Michel Marcus, Jun 29 2019

2*sinh(Pi) \\ Charles R Greathouse IV, Jan 30 2025

Equals A039661 - A093580.

Equals 2*sinh(Pi).

A194555 Decimal expansion of the real part of i^(e^Pi), where i = sqrt(-1).

Original entry on oeis.org

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

Offset: 0

Jonathan Sondow, Aug 28 2011

If Schanuel's Conjecture is true, then i^e^Pi is transcendental (see Marques and Sondow 2010, p. 79).

			i^e^Pi = 0.2192048949... - 0.9756788478...*i

G. C. Greubel, Table of n, a(n) for n = 0..10000
Steven Finch, Errata and Addenda to Mathematical Constants, Jun 23 2012, Section 1.1
D. Marques and Jonathan Sondow, Schanuel's conjecture and algebraic powers z^w and w^z with z and w transcendental, arXiv:1010.6216 [math.NT], 2010-2011; East-West J. Math., 12 (2010), 75-84.
Wikipedia, Schanuel's conjecture

Cf. A039661 (e^Pi), A194554 (imaginary part).

Cf. A194348 (sqrt(2)^(sqrt(2)^sqrt(2))).

RealDigits[ Re[I^E^Pi], 10, 100] // First

real(I^(exp(Pi))) \\ Michel Marcus, Aug 19 2018

cp's OEIS Frontend

A063503 Continued fraction for e^Pi - Pi^e (A063504 = A039661 - A059850).

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A042972 Decimal expansion of i^(-i), where i = sqrt(-1).

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A073244 Decimal expansion of Pi - e.

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A093580 Decimal expansion of e^(-Pi).

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A063504 Decimal expansion of e^Pi - Pi^e.

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A058287 Continued fraction for e^Pi.

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A175314 Decimal expansion of exp(Pi) + exp(-Pi).

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