A073743 -id:A073743 - cp's OEIS frontend

A137204 Decimal expansion of e + 1/e.

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

Offset: 1

Mohammad K. Azarian, Mar 04 2008

			3.086161269630487556955811241514123365203058224731727...

Index entries for transcendental numbers.

Cf. A001113, A068985, A073743.

RealDigits[E + 1/E, 10, 120][[1]] (* Amiram Eldar, Jun 20 2023 *)

exp(1)+1/exp(1) \\ Michel Marcus, Mar 14 2013

Equals 2*A073743. - Bruno Berselli, Mar 14 2013

A143821 Decimal expansion of the constant 1/2! + 1/5! + 1/8! + ... = 0.50835 81599 84216 ... .

Original entry on oeis.org

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

Offset: 0

Peter Bala, Sep 03 2008

Define a sequence of real numbers R(n) by R(n) := Sum_{k >= 0} (3*k)^n/(3*k)! for n = 0,1,2... . This constant is R(1); the decimal expansions of R(0) = 1 + 1/3!+ 1/6! + 1/9! + ... and R(2) - R(1) = 1/1! + 1/4! + 1/7! + ... may be found in A143819 and A143820. It is easy to verify that the sequence R(n) satisfies the recurrence relation u(n+3) = 3*u(n+2) - 2*u(n+1) + Sum_{i = 0..n} binomial(n,i) *3^(n-i)*u(i). Hence R(n) is an integral linear combination of R(0), R(1) and R(2) and so also an integral linear combination of R(0), R(1) and R(2) - R(1). Some examples are given below.

			R(n) as a linear combination of R(0), R(1) and R(2) - R(1).
=======================================
..R(n)..|.....R(0).....R(1)...R(2)-R(1)
=======================================
..R(3)..|.......1........1........3....
..R(4)..|.......6........2........7....
..R(5)..|......25.......11.......16....
..R(6)..|......91.......66.......46....
..R(7)..|.....322......352......203....
..R(8)..|....1232.....1730.....1178....
..R(9)..|....5672.....8233.....7242....
..R(10).|...32202....39987....43786....
...
The column entries are from A143815, A143816 and A143817.

Michael I. Shamos, A catalog of the real numbers, (2011). See p. 460.

Cf. A073742, A073743, A143815, A143816, A143817, A143818, A143819, A143820, A346441.

RealDigits[ N[ -((Cos[Sqrt[3]/2] - E^(3/2) + Sqrt[3]*Sin[Sqrt[3]/2])/(3*Sqrt[E])), 105]][[1]] (* Jean-François Alcover, Nov 08 2012 *)

Constant = (exp(1) + w*exp(w) + w^2*exp(w^2))/3, where w = exp(2*Pi*i/3). A143819 + A143820 + A143821 = exp(1).

Continued fraction: 1/(2 - 2/(61 - 60/(337 - 336/(991 - ... - P(n-1)/((P(n) + 1) - ... ))))), where P(n) = (3*n)*(3*n + 1)*(3*n + 2) for n >= 1. Cf. A346441. - Peter Bala, Feb 22 2024

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

A073745 Decimal expansion of csch(1).

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Aug 07 2002

csch(x) = 2/(e^x - e^(-x)).

By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 14 2019

			0.85091812823932154513384276328...

Samuel M. Selby (ed.), CRC Basic Mathematical Tables, CRC Press, 1970, p. 218.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Hyperbolic Cosecant.
Eric Weisstein's World of Mathematics, Hyperbolic Functions.
Index entries for transcendental numbers

Cf. A068139 (continued fraction), A073742 (sinh(1)=1/A073745), A073743 (cosh(1)), A073744 (tanh(1)), A073746 (sech(1)), A073747 (coth(1)).

RealDigits[Csch[1], 10, 100][[1]] (* Amiram Eldar, May 15 2021 *)

PARI
```
1/sinh(1)
```

Equals Sum_{k>=0} B(2*k) * (2 - 2^(2*k)) / (2*k)!, where B(k) is the k-th Bernoulli number. - Amiram Eldar, May 15 2021

A068118 Continued fraction expansion for cosh(1).

Original entry on oeis.org

1, 1, 1, 5, 3, 3, 2, 1, 21, 1, 1, 1, 4, 2, 1, 48, 71, 7, 1, 1, 9, 1, 2, 1, 11, 1, 5, 1, 2, 3, 2, 2, 2, 1, 1, 2, 10, 1, 1, 5, 26, 1, 25, 1, 2, 1, 5, 1, 2, 2, 7, 1, 1, 8, 8, 2, 1, 7, 2, 1, 9, 1, 3, 1, 4, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 5, 1, 1, 3, 1, 1, 1, 3, 7, 183

Offset: 0

Benoit Cloitre, Mar 13 2002

If an extra zero is added to the beginning of this sequence, continued fraction for sech(1) = 1/cosh(1). - Rick L. Shepherd, Aug 07 2002

Cf. A068139, A073743 (decimal expansion), A073746 (decimal expansion of sech(1)).

Cf. A078982, A078983 (convergents).

ContinuedFraction[Cosh[1],90] (* Harvey P. Dale, Apr 29 2013 *)

Offset changed by Andrew Howroyd, Aug 05 2024

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

A354334 a(n) is the numerator of Sum_{k=0..n} 1 / (2*k)!.

Original entry on oeis.org

1, 3, 37, 1111, 6913, 799933, 739138093, 44841044309, 32285551902481, 9879378882159187, 1251387991740163687, 1734423756551866870183, 136771701945232930334431, 23048564587067030852654113, 42769754577382930342215977687, 409306551305554643375006906464591

Offset: 0

Ilya Gutkovskiy, May 24 2022

			1, 3/2, 37/24, 1111/720, 6913/4480, 799933/518400, 739138093/479001600, ...

Cf. A010050, A053557, A061354, A073743, A103816, A120265, A143382, A354211, A354332, A354335 (denominators).

Table[Sum[1/(2 k)!, {k, 0, n}], {n, 0, 15}] // Numerator
nmax = 15; CoefficientList[Series[Cosh[Sqrt[x]]/(1 - x), {x, 0, nmax}], x] // Numerator
Accumulate[1/(2*Range[0,20])!]//Numerator (* Harvey P. Dale, Sep 05 2024 *)

a(n) = numerator(sum(k=0, n, 1/(2*k)!)); \\ Michel Marcus, May 24 2022

from fractions import Fraction
from math import factorial
def A354334(n): return sum(Fraction(1,factorial(2*k)) for k in range(n+1)).numerator # Chai Wah Wu, May 24 2022

Numerators of coefficients in expansion of cosh(sqrt(x)) / (1 - x).

A354335 a(n) is the denominator of Sum_{k=0..n} 1 / (2*k)!.

Original entry on oeis.org

1, 2, 24, 720, 4480, 518400, 479001600, 29059430400, 20922789888000, 6402373705728000, 810967336058880000, 1124000727777607680000, 88635485961891348480000, 14936720782466875392000000, 27717122237428532772864000000, 265252859812191058636308480000000

Offset: 0

Ilya Gutkovskiy, May 24 2022

			1, 3/2, 37/24, 1111/720, 6913/4480, 799933/518400, 739138093/479001600, ...

Cf. A010050, A053556, A061355, A073743, A143383, A354331, A354333, A354334 (numerators).

Table[Sum[1/(2 k)!, {k, 0, n}], {n, 0, 15}] // Denominator
nmax = 15; CoefficientList[Series[Cosh[Sqrt[x]]/(1 - x), {x, 0, nmax}], x] // Denominator

a(n) = denominator(sum(k=0, n, 1/(2*k)!)); \\ Michel Marcus, May 24 2022

from fractions import Fraction
from math import factorial
def A354335(n): return sum(Fraction(1,factorial(2*k)) for k in range(n+1)).denominator # Chai Wah Wu, May 24 2022

Denominators of coefficients in expansion of cosh(sqrt(x)) / (1 - x).

A052591 Expansion of e.g.f. x/((1-x)(1-x^2)).

Original entry on oeis.org

0, 1, 2, 12, 48, 360, 2160, 20160, 161280, 1814400, 18144000, 239500800, 2874009600, 43589145600, 610248038400, 10461394944000, 167382319104000, 3201186852864000, 57621363351552000, 1216451004088320000, 24329020081766400000, 562000363888803840000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Stirling transform of 2*a(n) = [2,4,24,96,...] is A052841(n+1) = [2,6,38,270,...]. - Michael Somos, Mar 04 2004

a(n) is the number of even fixed points in all permutations of {1,2,...,n+1}. Example: a(2)=2 because we have 12'3, 132, 312, 213, 231, and 32'1, the even fixed points being marked. - Emeric Deutsch, Jul 18 2009

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 536.

Cf. A008619, A052841.
Cf. A052558. - Emeric Deutsch, Jul 18 2009
Cf. A001620, A073743, A099284.

spec := [S,{S=Prod(Z,Sequence(Z),Sequence(Prod(Z,Z)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
G(x):=x/(1-x)/(1-x^2): f[0]:=G(x): for n from 1 to 19 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..19); # Zerinvary Lajos, Apr 03 2009

a(n)=if(n<0,0,n!*polcoeff(x/(1-x)/(1-x^2)+x*O(x^n),n))

Recurrence: {a(1)=1, a(0)=0, (-n^3 - 5*n^2 - 8*n - 4)*a(n) + (-2-n)*a(n+1) + (n+1)*a(n+2) = 0}.
a(n) = ((1/4)*(-1)^(1-n) + (1/2)*n + 1/4)*n!.
E.g.f.: x/((1-x)*(1-x^2)).
From Emeric Deutsch, Jul 18 2009: (Start)
a(n) = (n+1)!/2 if n is odd; a(n) = n!*n/2 if n is even.
a(n) = (n+1)! - A052558(n). (End)
a(n) = n!*A008619(n-1), n > 1. - R. J. Mathar, Nov 27 2011
Sum_{n>=1} 1/a(n) = 2*(CoshIntegral(1) + cosh(1) - gamma - 1) = 2*(A099284 + A073743 - A001620 - 1). - Amiram Eldar, Jan 22 2023

A080049 Operation count to create all permutations of n distinct elements using Algorithm L (lexicographic permutation generation) from Knuth's The Art of Computer Programming, Vol. 4, chapter 7.2.1.2. Sequence gives number of interchange operations in step L4.

Original entry on oeis.org

0, 2, 11, 63, 388, 2734, 21893, 197069, 1970726, 21678036, 260136487, 3381774403, 47344841720, 710172625898, 11362762014473, 193166954246169, 3477005176431178, 66063098352192544, 1321261967043851051, 27746501307920872271, 610423028774259190172, 14039729661807961374198

Offset: 2

Hugo Pfoertner, Jan 24 2003

Donald E. Knuth: The Art of Computer Programming, Volume 4, Fascicle 2, Generating All Tuples and Permutations. Addison-Wesley (2005). Chapter 7.2.1.2, 39-40.

D. E. Knuth, TAOCP Vol. 4, Pre-fascicle 2b (generating all permutations).
R. J. Ord-Smith, Generation of permutation sequences: Part 1, Computer J., 13 (1970), 151-155.
Hugo Pfoertner, FORTRAN implementation of Knuth's Algorithm L for lexicographic permutation generation.

Cf. A080047, A080048, A038155, A038156, A056542, A073743, A079756, A080047, A080048.

c FORTRAN program available at Pfoertner link.

a(2)=0, a(n)=n*a(n-1) + (n-1)*floor((n-1)/2).
c = limit n ->infinity a(n)/n! = 0.5430806.. = (e+1/e)/2-1 = A073743 - 1.
a(n) = floor (c*n! - (n-1)/2) for n>=2.

A109231 a(n) = floor(n*cosh(1)).

Original entry on oeis.org

1, 3, 4, 6, 7, 9, 10, 12, 13, 15, 16, 18, 20, 21, 23, 24, 26, 27, 29, 30, 32, 33, 35, 37, 38, 40, 41, 43, 44, 46, 47, 49, 50, 52, 54, 55, 57, 58, 60, 61, 63, 64, 66, 67, 69, 70, 72, 74, 75, 77, 78, 80, 81, 83, 84, 86, 87, 89, 91, 92, 94, 95, 97, 98, 100

Offset: 1

Reinhard Zumkeller, Jun 23 2005

nonn

Beatty sequence for cosh(1) = (e+1/e)/2 = 1.54308...= A073743; complement of A109232.

Eric Weisstein's World of Mathematics, Hyperbolic Cosine
Eric Weisstein's World of Mathematics, Beatty Sequence
Index entries for sequences related to Beatty sequences

Cf. A073743, A109232, A109233, A109234, A109237, A059563.

cp's OEIS Frontend

A137204 Decimal expansion of e + 1/e.

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A143821 Decimal expansion of the constant 1/2! + 1/5! + 1/8! + ... = 0.50835 81599 84216 ... .

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A073745 Decimal expansion of csch(1).

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A068118 Continued fraction expansion for cosh(1).

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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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A354334 a(n) is the numerator of Sum_{k=0..n} 1 / (2*k)!.

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A354335 a(n) is the denominator of Sum_{k=0..n} 1 / (2*k)!.

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A052591 Expansion of e.g.f. x/((1-x)(1-x^2)).

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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.