A083124 -id:A083124 - cp's OEIS frontend

A060402 Continued fraction expansion of tanh(Pi).

0, 1, 267, 4, 14, 1, 2, 1, 2, 2, 1, 2, 3, 8, 3, 1, 5, 4, 1, 14, 10, 1, 20, 38, 1, 2, 7, 5, 2, 1, 10, 1, 6, 1, 6, 1, 2, 3, 2, 1, 7, 1, 1, 5, 12, 2, 1, 1, 1, 2, 1, 1, 1, 34, 1, 8, 1, 33, 1, 4, 7, 1, 2, 56, 1, 3, 1, 34, 9, 1, 1, 7, 1, 3, 1, 7, 1, 4, 3, 1, 2, 14, 1, 10, 2, 51, 1, 6, 7, 17, 1, 14, 1, 8, 1, 1

Offset: 0

Bill Gosper, Apr 04 2001

			0.996272076220749944264690... = 0 + 1/(1 + 1/(267 + 1/(4 + 1/(14 + ...)))). - _Harry J. Smith_, Jul 04 2009

J. M. Borwein, D. H. Bailey and R. Girgensohn, Experimentation in Mathematics, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See p. 11.

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

Cf. A083124, A190352.

with(numtheory): c := cfrac (tanh(Pi),300,'quotients');

ContinuedFraction[Tanh[Pi], 100] (* Paolo Xausa, Jul 04 2024 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(tanh(Pi)); for (n=0, 20000, write("b060402.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jul 04 2009

More terms from James Sellers, Apr 06 2001

A084304 Continued fraction expansion of tanh(Pi^2).

Original entry on oeis.org

0, 1, 186895766, 8, 1, 11, 2, 3, 5, 8, 2, 2, 1, 19, 2, 23, 3, 1, 3, 1, 4, 2, 1, 1, 1, 2, 2, 25, 1, 19, 1, 1, 1, 6, 6, 2, 11, 1, 5, 2, 11, 1, 3, 1, 2, 1, 5, 1, 2, 7, 15, 4, 1, 7, 24, 1, 1, 1, 18, 2, 2, 25, 6, 4, 1, 1, 1, 2, 1, 2, 2, 2, 40, 1, 3, 3, 2, 2, 2, 16, 1, 361, 1, 2, 1, 4, 1, 13, 1, 5, 2, 1, 5

Offset: 0

Labos Elemer, Jun 04 2003

The least term that is larger than a(2) is a(6427449) = 890522226. - Amiram Eldar, Mar 08 2025

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A060402, A083124.

with(numtheory): c := cfrac (tanh(Pi^2), 300, 'quotients');

ContinuedFraction[Tanh[Pi^2],120] (* Harvey P. Dale, Nov 10 2021 *)

contfrac(tanh(Pi^2)) \\ Michel Marcus, Apr 05 2015

A367960 Decimal expansion of tanh(Pi/2).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Dec 06 2023

			0.91715233566727434637309...

Calvin C. Clawson, Mathematical Mysteries: The Beauty and Magic of Numbers, Springer, 2013. See p. 225.

Cf. A367961, A367959, A308715, A083124 (cont. frac).

Maple
```
evalf(tanh(Pi/2)) ;
```

First[RealDigits[Tanh[Pi/2],10,100]] (* Paolo Xausa, Dec 06 2023 *)

Equals 1/A367961 = A367959 / A308715 = (2/Pi)*A228048.

Equals (e^Pi - 1)/(e^Pi + 1) = K_{n>0} Pi^(2-[n=1])/(4*n - 2) (see Clawson at p. 225). - Stefano Spezia, Jul 01 2024

A084305 Continued fraction expansion of tanh(tanh(1)).

Original entry on oeis.org

0, 1, 1, 1, 3, 1, 5, 3, 1, 2, 85, 1, 2, 2, 2, 1, 1, 1, 2, 5, 1, 7, 1, 1, 4, 1, 4, 1, 1, 1, 1, 6, 3, 1, 3, 3, 2, 1, 9, 12, 16, 4, 1, 1, 1, 2, 1, 2, 1, 49, 3, 7, 1, 2, 6, 1, 1, 1, 6, 1, 3, 2, 1, 1, 5, 2, 1, 199, 8, 1, 8, 1, 1, 2, 2, 1, 9, 1, 4, 4, 4, 2, 2, 1, 2, 1, 3, 1, 249, 1, 2, 2, 1, 1, 4, 1, 1, 135, 3, 2, 1

Offset: 0

Labos Elemer, Jun 04 2003

While the continued fraction of tanh(1) = {0, 1, 3, 5, 7, 11, ..}, i.e., 0 and odd numbers, twice applying tanh() we get this sequence.

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A083124, A084304.

with(numtheory); cfrac(tanh(tanh(tanh(1))),300,'quotients');

ContinuedFraction[Tanh[Tanh[1]],120] (* Harvey P. Dale, Jan 19 2024 *)

contfrac(tanh(tanh(1))) \\ Amiram Eldar, Mar 08 2025

cp's OEIS Frontend

A060402 Continued fraction expansion of tanh(Pi).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A084304 Continued fraction expansion of tanh(Pi^2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A367960 Decimal expansion of tanh(Pi/2).

Views

Author

Keywords

Examples

References

Crossrefs

Programs

Maple

Mathematica

Formula

A084305 Continued fraction expansion of tanh(tanh(1)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI