A019609 -id:A019609 - cp's OEIS frontend

A096408 Decimal expansion of 1/(Pi*e).

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

Offset: 0

Mohammad K. Azarian, Aug 07 2004

			0.1170996630486383213804845369333337444

Cf. A000796, A001113, A019609.

RealDigits[1/(Pi E),10,120][[1]] (* Harvey P. Dale, Jul 22 2012 *)

Definition corrected by N. J. A. Sloane, Nov 17 2006

A096409 Decimal expansion of (Pi*e)^(-2).

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Aug 07 2004

			0.0137123310861046310874465544...

Muniru A Asiru, Table of n, a(n) for n = 0..2000

Cf. A000796, A001113, A019609, A092036.

evalf(1/(Pi^2*exp(2))) ; # R. J. Mathar, Oct 03 2011

Join[{0},RealDigits[1/(Pi E)^2,10,120][[1]]] (* Harvey P. Dale, May 05 2022 *)

(Pi*exp(1))^-2 \\ Patrick C. Schneider, Jun 27 2018

A121283 a(n) = floor(nPie).

Original entry on oeis.org

0, 8, 17, 25, 34, 42, 51, 59, 68, 76, 85, 93, 102, 111, 119, 128, 136, 145, 153, 162, 170, 179, 187, 196, 204, 213, 222, 230, 239, 247, 256, 264, 273, 281, 290, 298, 307, 315, 324, 333, 341, 350, 358, 367, 375, 384, 392, 401, 409, 418, 426, 435, 444, 452, 461

Offset: 0

Mohammad K. Azarian, Aug 24 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A019609, A032739, A062541.

Table[Floor[n Pi E], {n, 0, 80}] (* Vincenzo Librandi, Feb 22 2013 *)

Offset corrected by Mohammad K. Azarian, Nov 20 2008

Changed offset from 1 to 0 by Vincenzo Librandi, Feb 22 2013

A287596 a(n) is the denominator of r(n), where r(n) = r(n-1) + r(n-2)/(2*(n-1)) with r(0) = 0, r(1) = 1.

Original entry on oeis.org

1, 1, 1, 4, 12, 96, 480, 5760, 8064, 645120, 5806080, 116121600, 1277337600, 6131220480, 398529331200, 11158821273600, 167382319104000, 5356234211328000, 18211196318515200, 3278015337332736000, 62282291409321984000, 146546568021934080000, 52317124783830466560000

Offset: 0

Terry D. Grant, May 27 2017

The numerators are in A286307.
From Wolfdieter Lang, Jun 07 2017: (Start)
According to a Benoit Cloitre Aug 14 2003 formula in A019609 lim_{n-> oo} 4*n/r(n-1)^2 = Pi*e.
r(n+1) seems to be A268363(n) = 2^floor(n/2) * n!, n >= 0, up to n = 7, 12, 17, 20, 22, 27, 31, 32, 34,... (End)

Cf. A019609, A268363, A286307.

Denominator[RecurrenceTable[{r[n] == r[n - 1] + r[n - 2]/(2 (n - 1)), r[0] == 0, r[1] == 1}, r, {n, 0, 22}]]

a(n) = if(n < 2, return(n)); n++; my(v=vector(n)); v[1]=0; v[2] = 1; for(i = 3, n, v[i] = v[i-1] + v[i-2]/(2*i - 4)); denominator(v[#v])

a(n) = denominator(r(n)), where r(n) = r(n-1) + r(n-2)/2*(n-2) with r(0)=0 and r(1)=1.

G.f. of {r(n)}{n>=0}: x*exp(-x/2)/(1-x)^(3/2). - _Wolfdieter Lang, Jun 07 2017

A335027 Decimal expansion of Pi*(e-1)/2.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, May 20 2020

The value of an integral (see formula) first calculated by Cauchy in 1825 (with an error that was corrected in 1826).

This integral appears in the forward to Vălean's book, written by Paul J. Nahin.

			2.69907078454188691350045374313353580541885956819500...

Augustin-Louis Cauchy, Mémoire sur les intégrales définies prises entre des limites imaginaires, Paris, 1825, p. 65, equation 24.
Augustin-Louis Cauchy, Exercices de mathématiques, Paris, 1826, p. 108, equation 48.
Edward Thomas Copson, An Introduction to the Theory of Functions of a Complex Variable, London, 1935, Oxford, 1972 edition, p. 153.
Paul J. Nahin, Inside Interesting Integrals, Springer-Verlag New York, 2015, p. 340.
Cornel Ioan Vălean, (Almost) Impossible Integrals, Sums, and Series, Springer International Publishing, 2019, p. 206-207.

Cf. A000796 (Pi), A001113 (e), A019609 (Pi*e), A019610(Pi*e/2), A019669 (Pi/2), A335028.

Mathematica
```
RealDigits[Pi*(E-1)/2, 10, 100][[1]]
```

Pi*(exp(1)-1)/2 \\ Michel Marcus, May 20 2020

Equals Integral_{x=0..oo} (exp(cos(x)) * sin(sin(x))/x) * dx (Cauchy, 1825-26).
Equals Integral_{x=0..oo} (exp(cos(x)) * sin(x) * sin(sin(x))/x^2) * dx (Vălean, 2019).
Equals A019610 - A019669.

A335028 Decimal expansion of Pi*(exp(1/e) - 1)/2.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, May 20 2020

The value of an integral (see formula) first calculated by Cauchy in 1825 (with an error that was corrected in 1826).

			0.69848264271788427226723035849771244456284836693292...

Harold P. Boas, Cauchy’s Residue Sore Thumb, The American Mathematical Monthly, Vol. 125, No. 1 (2018), pp. 16-28, preprint, arXiv:1701.04887v1 [math.HO], 2017.
Augustin-Louis Cauchy, Mémoire sur les intégrales définies prises entre des limites imaginaires, Paris, 1825, p. 65, equation 24.
Augustin-Louis Cauchy, Exercices de mathématiques, Paris, 1826, p. 108, equation 48.

Cf. A000796 (Pi), A001113 (e), A019609 (Pi*e), A019610(Pi*e/2), A073229 (e^(1/e)), A335027.

RealDigits[Pi*(Exp[1/E] - 1)/2, 10, 100][[1]]

Pi*(exp(1/exp(1)) - 1)/2 \\ Michel Marcus, May 20 2020

Equals Integral_{x=0..oo} (exp(cos(x)) * sin(sin(x)) * x /(x^2 + 1)) * dx.

A352396 Integer part of e[n]Pi, where [n] indicates hyper-n, e = 2.718281828459045..., and Pi = 3.141592653589793... (using H. Kneser's proposal for n > 3).

Original entry on oeis.org

4, 5, 8, 23, 37149801960

Offset: 0

Marco Ripà, Apr 08 2022

The first term of this sequence is given by floor(e[0]Pi) = floor(Pi + 1) = floor(4.14159) = 4, which is the integer part of "e zeration Pi". In general, zeration is not a commutative arithmetic operation, while floor(e[1]Pi) = floor(Pi + e) = floor(5.85987) = 5 and floor(e[2]Pi) = floor(Pi * e) = floor(8.53973) = 8 hold since e[1]Pi = Pi[1]e and e[2]Pi = Pi[2]e.
If n = 3, then floor(e[3]Pi) = floor(e^Pi) = floor(23.14069) = 23 (if n > 2, then hyper-n is not characterized by the commutative property anymore, even if we can find fascinating examples as 4[3]2 = 2[3]4 = 16).
Now, tetration can be extended to complex bases as described in the Paulsen reference and the corresponding term of the present sequence can be found using his online calculator (see Links), so we have that floor(e[4]Pi) = floor(37149801960.55) = 37149801960. An easy proof that 37149801960.55999 > e^^Pi > 37149801960.55 follows from the chain of inequalities 37149801960.5569855999 > |37149801960.5569855 + 5.9249049902894650649*10^(-11)| > e^^Pi > |37149801960.556985498 + 5.9249049902894650647*10^(-11)| > 37149801960.55.
As far as we know, it has not been proved if e^^Pi is an irrational number (or not).

			For n = 3, a(3) = floor(e[3]Pi) = floor(e^Pi) = 15.

Hellmuth Kneser, Reelle analytische Lösungen der Gleichung phi(phi(x)) = e^x und verwandter Funktionalgleichungen, J. reine angew. Math. 187, 56-67 (1950)
Sheldon Levenstein (user sheldonison), New fatou.gp program, Jul 10 2015, updated Aug 14 2019.
William Paulsen, Tetration.
William Paulsen, Tetration for complex bases, Advances in Computational Mathematics, Vol. 45, No. 1 (2019), pp. 243-267; ResearchGate link.
Wikipedia, Hyperoperation
Wikipedia, Tetration

Cf. A000796, A001113, A019609, A039661, A059742, A351727, A353246.

a(n) = floor(e[n]Pi).

A092138 Decimal expansion of (Pi*e)^3.

Original entry on oeis.org

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

Offset: 3

Mohammad K. Azarian, Mar 30 2004

			622.777715112...

Cf. A019609 (Pi*e), A092036 ((Pi*e)^2).

RealDigits[N[(Pi*E)^3,6! ]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jun 18 2009 *)

A092140 Decimal expansion of (Pi*e)^4.

Original entry on oeis.org

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

Offset: 4

Mohammad K. Azarian, Mar 30 2004

			5318.3561668666...

Cf. A019609 (Pi*e), A092036 ((Pi*e)^2).

RealDigits[N[(Pi*E)^4,6! ]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jun 18 2009 *)

cp's OEIS Frontend

A019632 Decimal expansion of Pi*e/24.

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Magma

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A096408 Decimal expansion of 1/(Pi*e).

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A096409 Decimal expansion of (Pi*e)^(-2).

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A121283 a(n) = floor(n*Pi*e).

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A287596 a(n) is the denominator of r(n), where r(n) = r(n-1) + r(n-2)/(2*(n-1)) with r(0) = 0, r(1) = 1.

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A335027 Decimal expansion of Pi*(e-1)/2.

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A335028 Decimal expansion of Pi*(exp(1/e) - 1)/2.

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A352396 Integer part of e[n]Pi, where [n] indicates hyper-n, e = 2.718281828459045..., and Pi = 3.141592653589793... (using H. Kneser's proposal for n > 3).

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A121283 a(n) = floor(nPie).