A097665 -id:A097665 - cp's OEIS frontend

A049006 Decimal expansion of i^i = exp(-Pi/2).

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

Offset: 0

Deepak R. N (deepak_rama(AT)bigfoot.com)

Equals 1/A042972. - Lekraj Beedassy, Sep 02 2005
Euler knew this number to be purely real, and called the fact "remarkable" in a letter to Goldbach dated June 14, 1746. - Alonso del Arte, Nov 30 2012
The value follows immediately from Euler's formula i = exp(i Pi/2) and the rule (a^b)^c = a^(b*c). - The value given by Uhler has the final digits ...14 instead ...08, which is compatible with the claimed accuracy of 52 digits. - M. F. Hasler, May 17 2018

			0.20787957635076190854695561983497877003387...

Florian Cajori, History of Mathematics. New York: Chelsea Publishing Company for the American Mathematical Society (1991): 236.
Ian Connell, Modern Algebra: A Constructive Introduction. New York: Elsevier (1981) p. 363.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.11, p. 449.
Roger Penrose, "The Road to Reality, A complete guide to the Laws of the Universe", Jonathan Cape, London, 2004, page 97.
Reinhold Remmert, Theory of Complex Functions: Readings in Mathematics. New York: Springer-Verlag (1991): 162.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 26.

Harry J. Smith, Table of n, a(n) for n = 0..20000
Leonhard Euler, Letter to Christian Goldbach, Berlin, June 14 1746.
Simon Plouffe, exp(-Pi/2) also i**i to 10000 digits.
H. S. Uhler, On the numerical value of i^i, Amer. Math. Monthly, 28 (1921), 114-116.
Eric Weisstein's World of Mathematics, i.
Index entries for transcendental numbers.

Cf. A042972, A049007, A097665, A202501 (tetration).

Cf. A077589 and A077590 for i^i^i^...

Mathematica
```
RealDigits[Re[N[I^I, 100]]][[1]]
```

{ default(realprecision, 20080); x=10*exp(-Pi/2); for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b049006.txt", n, " ", d)); } \\ Harry J. Smith, Apr 28 2009, corrected May 19 2009

digits(exp(-Pi/2)\.1^default(realprecision))[^-1] \\ M. F. Hasler, May 17 2018

Equals 1/A042972 = 2*A097665. - Hugo Pfoertner, Aug 21 2024

A097679 E.g.f.: (1/(1-x^4))exp( 4Sum_{i>=0} x^(4i+1)/(4i+1) ) for an order-4 linear recurrence with varying coefficients.

Original entry on oeis.org

1, 4, 16, 64, 280, 1600, 12160, 102400, 880000, 8358400, 94720000, 1189888000, 15213952000, 204285952000, 3092697088000, 51351519232000, 869951500288000, 15148619579392000, 287722152460288000, 5927812334878720000

Offset: 0

Paul D. Hanna, Sep 01 2004

nonn

Lim_{n->inf} n*n!/a(n) = 4*c = 0.4157591527... where c = 4*exp(psi(1/4)+EulerGamma) = 0.1039397881...(A097665) and EulerGamma is the Euler-Mascheroni constant (A001620) and psi() is the Digamma function (see Mathworld link).

			The sequence {1, 4, 16/2!, 64/3!, 280/4!, 1600/5!, 12160/6!, 102400/7!,...} is generated by a recursion described by _Benoit Cloitre_'s generalized Euler-Gauss formula for the Gamma function (see Cloitre link).

Mohammad K. Azarian, Problem 1218, Pi Mu Epsilon Journal, Vol. 13, No. 2, Spring 2010, p. 116. Solution published in Vol. 13, No. 3, Fall 2010, pp. 183-185.
A. M. Odlyzko, Linear recurrences with varying coefficients, in Handbook of Combinatorics, Vol. 2, R. L. Graham, M. Grotschel and L. Lovasz, eds., Elsevier, Amsterdam, 1995, pp. 1135-1138.

Benoit Cloitre, On a generalization of Euler-Gauss formula for the Gamma function, preprint 2004.
Andrew Odlyzko, Asymptotic enumeration methods, in Handbook of Combinatorics, vol. 2, 1995, pp. 1063-1229.
Eric Weisstein's World of Mathematics, Digamma Function.

Cf. A097665, A097677, A097678, A097680, A097681, A097682.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!((1+x)/(1-x^4)/(1-x)*Exp(2*Arctan(x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 29 2018

Range[0, 20]! CoefficientList[ Series[ E^(4Sum[x^(4k + 1)/(4k + 1), {k, 0, 150}])/(1 - x^4), {x, 0, 20}], x] (* Robert G. Wilson v, Sep 03 2004 *)

{a(n)=n!*polcoeff(1/(1-x^4)*exp(4*sum(i=0,n,x^(4*i+1)/(4*i+1)))+x*O(x^n),n)}

a(n)=if(n<0,0,if(n==0,1,4*a(n-1)+if(n<4,0,n!/(n-4)!*a(n-4))))

For n>=4: a(n) = 4*a(n-1) + n!/(n-4)!*a(n-4); for n<4: a(n)=4^n.

E.g.f.: (1+x)/(1-x^4)/(1-x)*exp(2*atan(x)).

A097663 Decimal expansion of the constant 3*exp(psi(1/3) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

Original entry on oeis.org

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

Offset: 0

Paul D. Hanna, Aug 25 2004

This constant appears in Benoit Cloitre's generalized Euler-Gauss formula for the Gamma function (see Cloitre link) and is involved in the exact determination of asymptotic limits of certain order-3 linear recursions with varying coefficients (see A097677 for example).

			0.23311909318456411730537562326544289574460858702592456414096...

A. M. Odlyzko, Linear recurrences with varying coefficients, in Handbook of Combinatorics, Vol. 2, R. L. Graham, M. Grotschel and L. Lovasz, eds., Elsevier, Amsterdam, 1995, pp. 1135-1138.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Benoit Cloitre, On a generalization of Euler-Gauss formula for the Gamma function, preprint 2004.
Xavier Gourdon and Pascal Sebah, Introduction to the Gamma Function.
Andrew Odlyzko, Asymptotic enumeration methods, in Handbook of Combinatorics, vol. 2, 1995, pp. 1063-1229.

Cf. A001620, A047787, A097664, A097665-A097676.

SetDefaultRealField(RealField(100)); R:= RealField(); Exp(-Pi(R)/Sqrt(12))/Sqrt(3); // G. C. Greubel, Sep 07 2018

RealDigits[1/Sqrt[3]*E^(-Pi/Sqrt[12]), 10, 105][[1]] (* Robert G. Wilson v, Aug 28 2004 *)

PARI
```
3*exp(psi(1/3)+Euler)
```

Equals exp(-Pi/sqrt(12))/sqrt(3).

More terms from Robert G. Wilson v, Aug 28 2004

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

A097664 Decimal expansion of the constant 3*exp(psi(2/3) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

Original entry on oeis.org

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

Offset: 1

Paul D. Hanna, Aug 25 2004

This constant appears in Benoit Cloitre's generalized Euler-Gauss formula for the Gamma function (see Cloitre link) and is involved in the exact determination of asymptotic limits of certain order-3 linear recursions with varying coefficients (see A097678 for example).

			c = 1.42988430840123420566179042477513809656498236767564464887634...

A. M. Odlyzko, Linear recurrences with varying coefficients, in Handbook of Combinatorics, Vol. 2, R. L. Graham, M. Grotschel and L. Lovasz, eds., Elsevier, Amsterdam, 1995, pp. 1135-1138.

G. C. Greubel, Table of n, a(n) for n = 1..2500
Benoit Cloitre, On a generalization of Euler-Gauss formula for the Gamma function, preprint 2004.
Andrew Odlyzko, Asymptotic enumeration methods, in Handbook of Combinatorics, vol. 2, 1995, pp. 1063-1229.
Xavier Gourdon and Pascal Sebah, Introduction to the Gamma Function.

Cf. A097663, A097665-A097676.

SetDefaultRealField(RealField(100)); R:= RealField(); (1/Sqrt(3))*Exp(Pi(R)/Sqrt(12)); // G. C. Greubel, Aug 27 2018

RealDigits[1/Sqrt[3]*E^(Pi/Sqrt[12]), 10, 105][[1]] (* Robert G. Wilson v, Aug 28 2004 *)

PARI
```
3*exp(psi(2/3)+Euler)
```

c = 1/sqrt(3)*exp(Pi/sqrt(12)).

More terms from Robert G. Wilson v, Aug 28 2004

A097666 Decimal expansion of the constant 4*exp(psi(3/4) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

Original entry on oeis.org

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

Offset: 1

Paul D. Hanna, Aug 25 2004

This constant appears in Benoit Cloitre's generalized Euler-Gauss formula for the Gamma function (see Cloitre link) and is involved in the exact determination of asymptotic limits of certain order-4 linear recursions with varying coefficients (see A097679 for example).

			c = 2.40523869048267582773651783335191656319508543733226747001040...

A. M. Odlyzko, Linear recurrences with varying coefficients, in Handbook of Combinatorics, Vol. 2, R. L. Graham, M. Grotschel and L. Lovasz, eds., Elsevier, Amsterdam, 1995, pp. 1135-1138.

G. C. Greubel, Table of n, a(n) for n = 1..2500
Benoit Cloitre, On a generalization of Euler-Gauss formula for the Gamma function, preprint 2004.
Xavier Gourdon and Pascal Sebah, Introduction to the Gamma Function.
Andrew Odlyzko, Asymptotic enumeration methods, in Handbook of Combinatorics, vol. 2, 1995, pp. 1063-1229.

Cf. A097663-A097665, A097667-A097676.

SetDefaultRealField(RealField(100)); R:= RealField(); Exp(Pi(R)/2)/2; // G. C. Greubel, Sep 07 2018

RealDigits[1/2*E^(Pi/2), 10, 105][[1]] (* Robert G. Wilson v, Aug 27 2004 *)

PARI
```
4*exp(psi(3/4)+Euler)
```

c = 1/2*exp(Pi/2).

More terms from Robert G. Wilson v, Aug 27 2004

cp's OEIS Frontend

A049006 Decimal expansion of i^i = exp(-Pi/2).

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A097679 E.g.f.: (1/(1-x^4))*exp( 4*Sum_{i>=0} x^(4*i+1)/(4*i+1) ) for an order-4 linear recurrence with varying coefficients.

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A097663 Decimal expansion of the constant 3*exp(psi(1/3) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

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A097664 Decimal expansion of the constant 3*exp(psi(2/3) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A097666 Decimal expansion of the constant 4*exp(psi(3/4) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A097679 E.g.f.: (1/(1-x^4))exp( 4Sum_{i>=0} x^(4i+1)/(4i+1) ) for an order-4 linear recurrence with varying coefficients.