A073821 -id:A073821 - cp's OEIS frontend

A181051 Incorrect duplicate of A073821.

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

Offset: 0

Jonathan D. B. Hodgson, Oct 01 2010

dead

Previous name was: Decimal expansion of the constant whose continued fraction representation is [0; 2, 4, 6, 8, ..., 2*n, ...], i.e., 2/(4+6/(8+10/(12+...) using every even integer.

			0.42703859021672487086140473103956401461415197068651827...

A073747 Decimal expansion of coth(1).

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Aug 07 2002

coth(x) = (e^x + e^(-x))/(e^x - e^(-x)).
Because the continued fraction for coth(1) is all positive odd numbers in sequence, the second Mathematica program below also generates the sequence. - Harvey P. Dale, Oct 15 2011
By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 14 2019

			1.31303528549933130363616124693...

Samuel M. Selby, editor, CRC Basic Mathematical Tables, CRC Press, 1970, p. 218.

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Hideyuki Ohtsuka, Problem 11853, The American Mathematical Monthly, Vol. 122, No. 7 (2015), p. 700; A Hyperbolic Sine Series, Solutions to Problem 11853 by Tewodros Amdeberhan and Rituraj Nandan, ibid., Vol. 124, No. 5 (2017), p. 469.
Allen Stenger, Experimental math for Math Monthly problems, The American Mathematical Monthly, Vol. 124, No. 2 (2017), pp. 116-131; alternative link.
Eric Weisstein's World of Mathematics, Hyperbolic Cotangent.
Eric Weisstein's World of Mathematics, Hyperbolic Functions.
Index entries for transcendental numbers

Cf. A005408 (continued fraction: odd numbers), A073821 (continued fraction exp. is even numbers), A073744 (tanh(1)=1/A073747), A073742 (sinh(1)), A073743 (cosh(1)), A073745 (csch(1)), A073746 (sech(1)), A349004.

RealDigits[Coth[1],10,120][[1]] (* or *) RealDigits[ FromContinuedFraction[ Range[1,1001,2]],10,120][[1]] (* Harvey P. Dale, Oct 15 2011 *) (* see Comments, above, for the second program *)

PARI
```
1/tanh(1)
```

Equals 1 + Sum_{n>=1} (2^(2*n)*B(2*n))/(2*n)! = 1 + Sum_{n>=1}  (-1)^(n+1)*2*(A046988(n+1) / A002432(n+1)). - Terry D. Grant, May 30 2017
Equals 1 + BesselI(3/2, 1)/BesselI(1/2, 1). - Terry D. Grant, Jun 18 2018
Equals 1 + Sum_{k>=1} csch(2^k) (Ohtsuka, 2015; Stenger, 2017). - Amiram Eldar, Oct 04 2021

A298242 Decimal expansion of BesselI(1,1/2)/BesselI(0,1/2).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Jan 15 2018

			0.2424996125808019453507023535036354074122660448659455966...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Continued Fraction Constants
Index entries for sequences related to Bessel functions or polynomials

Cf. A008586, A052119, A073744, A073747, A073821, A296168, A298241, A298243.

RealDigits[BesselI[1, 1/2]/BesselI[0, 1/2], 10, 110] [[1]]
RealDigits[Hypergeometric0F1[2, (1/2)^2/4]/(4 Gamma[2] Hypergeometric0F1[1, (1/2)^2/4]), 10, 110][[1]]

besseli(1,1/2)/besseli(0,1/2) \\ Michel Marcus, Jul 03 2018

Equals 1/(4 + 1/(8 + 1/(12 + 1/(16 + 1/(20 + 1/(24 + ...)))))).

A298241 Decimal expansion of BesselI(1,2/3)/BesselI(0,2/3).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Jan 15 2018

			0.3160892412682211840956016917105181147668629270070418207...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Continued Fraction Constants
Index entries for sequences related to Bessel functions or polynomials

Cf. A008585, A052119, A073744, A073747, A073821, A296168, A298242, A298243.

RealDigits[BesselI[1, 2/3]/BesselI[0, 2/3], 10, 110] [[1]]
RealDigits[Hypergeometric0F1[2, (2/3)^2/4] /(3 Gamma[2] Hypergeometric0F1[1, (2/3)^2/4]), 10, 110][[1]]

besseli(1,2/3)/besseli(0,2/3) \\ Michel Marcus, Jul 03 2018

Equals 1/(3 + 1/(6 + 1/(9 + 1/(12 + 1/(15 + 1/(18 + ...)))))).

A298243 Decimal expansion of BesselI(1,2/5)/BesselI(0,2/5).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Jan 15 2018

			0.1961038122179955134083610646268785173725058094464270021...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Continued Fraction Constants
Index entries for sequences related to Bessel functions or polynomials

Cf. A008587, A052119, A073744, A073747, A073821, A296168, A298241, A298242.

RealDigits[BesselI[1, 2/5]/BesselI[0, 2/5], 10, 110] [[1]]
RealDigits[Hypergeometric0F1[2, (2/5)^2/4]/(5 Gamma[2] Hypergeometric0F1[1, (2/5)^2/4]), 10, 110][[1]]

besseli(1,2/5)/besseli(0,2/5) \\ Michel Marcus, Jul 03 2018

Equals 1/(5 + 1/(10 + 1/(15 + 1/(20 + 1/(25 + 1/(30 + ...)))))).

A261827 Decimal expansion of the number whose continued fraction expansion consists of the perfect numbers (A000396).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Sep 02 2015

			6.0357117143069233346283990529260946180806175748136895461...

Eric Weisstein's World of Mathematics, Continued Fraction
Eric Weisstein's World of Mathematics, Perfect Number

Cf. A000396, A073821, A073822, A073823, A073824, A052119, A064442.

ind = {1, 2, 3, 4, 6, 7, 8, 11, 18, 24, 28, 31, 98, 111} (* from A016027 *); p = Prime@ ind; pn = (2^p - 1)(2^(p - 1)); RealDigits[ FromContinuedFraction@ pn, 10, 111][[1]] (* Robert G. Wilson v, Sep 13 2015 *)

cp's OEIS Frontend

A181051 Incorrect duplicate of A073821.

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A073747 Decimal expansion of coth(1).

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A298242 Decimal expansion of BesselI(1,1/2)/BesselI(0,1/2).

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A298241 Decimal expansion of BesselI(1,2/3)/BesselI(0,2/3).

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A298243 Decimal expansion of BesselI(1,2/5)/BesselI(0,2/5).

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A261827 Decimal expansion of the number whose continued fraction expansion consists of the perfect numbers (A000396).

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