A137204 -id:A137204 - cp's OEIS frontend

A073743 Decimal expansion of cosh(1).

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

Offset: 1

Rick L. Shepherd, Aug 07 2002

Also decimal expansion of cos(i). - N. J. A. Sloane, Feb 12 2010
cosh(x) = (e^x + e^(-x))/2.
Equals Sum_{n>=0} 1/A010050(n). See Gradsteyn-Ryzhik (0.245.5). - R. J. Mathar, Oct 27 2012
By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 14 2019

			1.54308063481524377847790562075...

S. Selby, editor, CRC Basic Mathematical Tables, CRC Press, 1970, p. 218.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 2, equation 2:5:6 at page 20.

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Hyperbolic Cosine.
Eric Weisstein's World of Mathematics, Hyperbolic Functions.
Eric Weisstein's World of Mathematics, Factorial Sums.
Index entries for transcendental numbers.

Cf. A068118 (continued fraction), A073742, A073744, A073745, A073746, A073747, A049470, A137204.

Digits:=100: evalf(cosh(1)); # Wesley Ivan Hurt, Nov 18 2014

RealDigits[Cosh[1],10,120][[1]] (* Harvey P. Dale, Aug 03 2014 *)

PARI
```
cosh(1)
```

Continued fraction representation: cosh(1) = 1 + 1/(2 - 2/(13 - 12/(31 - ... - (2*n - 4)*(2*n - 5)/((4*n^2 - 10*n + 7) - ... )))). See A051396 for proof. Cf. A049470 (cos(1)) and A073742 (sinh(1)). - Peter Bala, Sep 05 2016
Equals Product_{k>=0} 1 + 4/((2*k+1)*Pi)^2. - Amiram Eldar, Jul 16 2020
Equals 1/A073746 = A137204/2. - Hugo Pfoertner, Dec 27 2024

A174548 Decimal expansion of e - 1/e.

Original entry on oeis.org

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

Offset: 1

Paul Curtz, Mar 22 2010

			2.3504023872876029137647637...

Index entries for transcendental numbers

Cf. A001113 (e), A068985 (1/e), and A137204 (e + 1/e), A073742 (sinh(1)).

evalf(exp(1)-exp(-1)) ; # R. J. Mathar, Oct 14 2011

Mathematica
```
RealDigits[E - 1/E, 10, 111][[1]]
```

exp(1) - exp(-1) \\ Michel Marcus, May 05 2019

Equals 2 * sinh(1) = 2 * A073742. - Amiram Eldar, Nov 25 2020

Edited and extended by Robert G. Wilson v, Apr 25 2010

A059563 Beatty sequence for e + 1/e.

Original entry on oeis.org

3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 37, 40, 43, 46, 49, 52, 55, 58, 61, 64, 67, 70, 74, 77, 80, 83, 86, 89, 92, 95, 98, 101, 104, 108, 111, 114, 117, 120, 123, 126, 129, 132, 135, 138, 141, 145, 148, 151, 154, 157, 160, 163, 166, 169, 172, 175, 178, 182, 185

Offset: 1

Mitch Harris, Jan 22 2001

Harry J. Smith, Table of n, a(n) for n = 1..2000
Aviezri S. Fraenkel, Jonathan Levitt and Michael Shimshoni, Characterization of the set of values f(n)=[n alpha], n=1,2,..., Discrete Math. 2 (1972), no. 4, 335-345.
Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences

Beatty complement is A059564.

Cf. A137204 (e+1/e).

Floor[Range[100]*(E + 1/E)] (* Paolo Xausa, Jul 06 2024 *)

{ default(realprecision, 100); b=exp(1) + 1/exp(1); for (n = 1, 2000, write("b059563.txt", n, " ", floor(n*b)); ) } \\ Harry J. Smith, Jun 28 2009

a(n) = floor(n*(e+1/e)). - Michel Marcus, Jan 04 2015

A344262 a(0)=1; for n>0, a(n) = a(n-1)n+1 if n is even, (a(n-1)+1)n otherwise.

Original entry on oeis.org

1, 2, 5, 18, 73, 370, 2221, 15554, 124433, 1119906, 11199061, 123189682, 1478276185, 19217590418, 269046265853, 4035693987810, 64571103804961, 1097708764684354, 19758757764318373, 375416397522049106, 7508327950440982121, 157674886959260624562

Offset: 0

Amrit Awasthi, May 13 2021

nonn

			a(0) = 1;
a(1) = (a(0)+1)*1 =  (1+1)*1 =   2;
a(2) = (a(1)*2)+1 =  (2*2)+1 =   5;
a(3) = (a(2)+1)*3 =  (5+1)*3 =  18;
a(4) = (a(3)*4)+1 = (18*4)+1 =  73;
a(5) = (a(4)+1)*5 = (73+1)*5 = 370.

Cf. A033540, A073743, A090805, A137204, A155521, A344317.

a:= proc(n) a(n):= n*a(n-1) + n^(n mod 2) end: a(0):= 1:
seq(a(n), n=0..22);  # Alois P. Heinz, May 14 2021

a[1] = 1; a[n_] := a[n] = If[OddQ[n], (n - 1)*a[n - 1] + 1, (n - 1)*(a[n - 1] + 1)]; Array[a, 25] (* Amiram Eldar, May 13 2021 *)

E.g.f.: (x+1)*cosh(x)/(1-x). - Alois P. Heinz, May 14 2021
Lim_{n->infinity} a(n)/n! = 2*cosh(1) = A137204 = 2*A073743. - Amrit Awasthi, May 15 2021
a(n) = A344317(n) - A155521(n-1) for n > 0. - Alois P. Heinz, May 18 2021

A344419 a(n) = n*a(n-1) + n^(n mod 2), a(0) = 0.

Original entry on oeis.org

0, 1, 3, 12, 49, 250, 1501, 10514, 84113, 757026, 7570261, 83272882, 999274585, 12990569618, 181867974653, 2728019619810, 43648313916961, 742021336588354, 13356384058590373, 253771297113217106, 5075425942264342121, 106583944787551184562, 2344846785326126060365

Offset: 0

Alois P. Heinz, May 17 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A000142, A001113, A073743, A155521, A137204, A344262, A344418.

a:= proc(n) a(n):= n*a(n-1) + n^(n mod 2) end: a(0):= 0:
seq(a(n), n=0..23);

E.g.f.: ((x+1)*cosh(x)-1)/(1-x).
a(n) = A344262(n) - n! = A344262(n) - A000142(n).
a(n) = A344418(n) - A155521(n-1) for n > 0.
Lim_{n->infinity} a(n)/n! = 2*cosh(1)-1 = 2*A073743-1 = e+1/e-1 = A137204-1. - Amrit Awasthi, May 20 2021

A197220 Numerators of convergents to e + 1/e.

Original entry on oeis.org

3, 34, 37, 71, 108, 179, 1003, 1182, 12823, 39651, 369682, 409333, 1188348, 28929685, 4109203618, 12356540539, 16465744157, 61753773010, 263480836197, 325234609207, 2214888491439, 2540123100646, 14915503994669, 32371131089984, 79657766174637, 112028897264621

Offset: 1

T. D. Noe, Oct 12 2011

nonn

Robert Israel, Table of n, a(n) for n = 1..1898

Cf. A137204 (decimal), A197221 (denominators).

numtheory:-cfrac(exp(1)+exp(-1), 100, 'ccon'):
map(numer, ccon[1..-2]); # Robert Israel, Jun 08 2017

Mathematica
```
Numerator[Convergents[E + 1/E, 40]]
```

A197221 Denominator of convergents to e + 1/e.

Original entry on oeis.org

1, 11, 12, 23, 35, 58, 325, 383, 4155, 12848, 119787, 132635, 385057, 9374003, 1331493483, 4003854452, 5335347935, 20009898257, 85374940963, 105384839220, 717683976283, 823068815503, 4833028053798, 10489124923099, 25811277899996, 36300402823095

Offset: 1

T. D. Noe, Oct 12 2011

nonn

Robert Israel, Table of n, a(n) for n = 1..1899

Cf. A137204 (decimal), A197220 (numerators).

numtheory:-cfrac(exp(1)+exp(-1), 100, 'ccon'):
map(denom, ccon[1..-2]); # Robert Israel, Jun 08 2017

Mathematica
```
Denominator[Convergents[E + 1/E, 40]]
```

A196080 Numerators of the sum of the n-th partial sums of the expansions of e and 1/e.

Original entry on oeis.org

2, 2, 3, 3, 37, 37, 1111, 1111, 6913, 6913, 799933, 799933, 739138093, 739138093, 44841044309, 44841044309, 32285551902481, 32285551902481, 9879378882159187, 9879378882159187, 1251387991740163687

Offset: 0

Paul Curtz, Sep 27 2011

The n-th partial sums of the Taylor expansion of E are f(n) = A061354(n)/A061355(n) = 1, 2, 5/2, 8/3, 65/24, 163/60,.. .
The partial sums of an expansion of 1/e are essentially A000255(n-2)/A001048(n-1) preceded by 1 and 0, namely  g(n)= 1, 0, 1/2, 1/3, 3/8, 11/30, 53/144, 103/280, 2119/5760,... (Jolley's partial sums of 1/E in A068985 is the bisection 0, 1/3, 11/30, 103/280, 16687/45360,... of g(n).)
The current sequence are the numerators of f(n)+g(n), converging to E+1/E, namely 2, 2, 3, 3, 37/12, 37/12, 1111/360, 1111/360, 6913/2240 = 62217/21060, 6913/2240 = 62217/21060, 799933/259200 = 5599531/1814400,... The unreduced fractions are apparently given by duplicated A051396(n+1)/A002674(n).

			a(0)=1+1, a(1)=2+0, a(2)=(5+1)/2, a(3)=(8+1)/3.

Cf. A001113, A068985, A137204 (e+1/e).

a[n_] := (E*Gamma[n+1, 1] + (1/E)*Gamma[n+1, -1])/n! // FullSimplify // Numerator; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Aug 02 2012 *)

Redefined by reduced fractions. - R. J. Mathar, Jul 02 2012

cp's OEIS Frontend

A073743 Decimal expansion of cosh(1).

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A174548 Decimal expansion of e - 1/e.

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A059563 Beatty sequence for e + 1/e.

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A344262 a(0)=1; for n>0, a(n) = a(n-1)*n+1 if n is even, (a(n-1)+1)*n otherwise.

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A344419 a(n) = n*a(n-1) + n^(n mod 2), a(0) = 0.

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A197220 Numerators of convergents to e + 1/e.

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Maple

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A197221 Denominator of convergents to e + 1/e.

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Maple

Mathematica

A196080 Numerators of the sum of the n-th partial sums of the expansions of e and 1/e.

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A344262 a(0)=1; for n>0, a(n) = a(n-1)n+1 if n is even, (a(n-1)+1)n otherwise.