A019774 -id:A019774 - cp's OEIS frontend

A334367 Decimal expansion of Sum_{k>=0} 1/(4*k+2)!!.

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

Offset: 0

Ilya Gutkovskiy, Apr 24 2020

This constant is transcendental.

			1/(2^1*1!) + 1/(2^3*3!) + 1/(2^5*5!) + ... = 0.52109530549374736162242...

Index entries for transcendental numbers

Cf. A019774, A067626, A073742, A143280, A160327, A201504, A334366.

Mathematica
```
RealDigits[Sinh[1/2], 10, 110] [[1]]
```
PARI
```
sinh(1/2) \\ Michel Marcus, Apr 25 2020
```

Equals sinh(1/2).

Equals (1/2) * Product_{k>=1} 1 + 1/(2*k*Pi)^2. - Amiram Eldar, Jul 16 2020

A019778 Decimal expansion of sqrt(e)/5.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

			0.32974425414002562936973015756283271433075522014202...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index entries for transcendental numbers

Cf. A019774.

RealDigits[Sqrt[E]/5,10,120][[1]] (* Harvey P. Dale, Aug 25 2019 *)

Equals 0.1 * 2*sqrt(e) = 0.1 * Sum_{k>=0} (2*k+1)/(2^k * k!). - Amiram Eldar, Aug 07 2020

A092618 Decimal expansion of e^(-1/5).

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Apr 22 2004

			0.81873075307798

Cf. A001113, A019774, A068985.

RealDigits[Power[E, (-5)^-1],10,120][[1]] (* Harvey P. Dale, Jan 27 2012 *)

A092750 Decimal expansion of e^(-1/7).

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Apr 22 2004

			0.866877899750...

Cf. A001113, A019774, A068985.

evalf(1/exp(1/7), 124);  # Alois P. Heinz, Apr 04 2020

RealDigits[N[Exp[-1/7], 112]][[1]] (* Georg Fischer, Apr 04 2020 *)

a(104) corrected by Georg Fischer, Apr 04 2020

A181180 Decimal expansion of exp(exp(-1/2)).

Original entry on oeis.org

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

Offset: 1

Geoffrey Caveney, Oct 09 2010

The real number y such that y = exp(xy^-x) is a maximum, at x=exp(1/2).

y=1.8340573791984...

Cf. A092605 (decimal expansion of exp(-1/2)), A019774 (decimal expansion of exp(1/2)). - Klaus Brockhaus, Oct 09 2010

RealDigits[ E^E^(-1/2), 10, 111][[1]] (* Robert G. Wilson v, Oct 09 2010 *)

More terms from Klaus Brockhaus, Oct 09 2010

A227569 Decimal expansion of maximal value of function F[a(n); b(n)] for pairs of complements a(n) and b(n) of natural numbers A000027, where a(n) = odd numbers (A005408) and b(n) = even numbers (A005843); see Comments for the definition of function F[a(n); b(n)].

Original entry on oeis.org

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

Offset: 1

Jaroslav Krizek, Jul 16 2013

Apart from the first digit, the same as A143280. The sum of the reciprocals of the double factorial numbers, Sum_{n>=1} 1/n!! = Sum_{n>=2} n!!/n!. - Robert G. Wilson v, Jun 27 2015
Definition of function F[a(n); b(n)]: Let a(n) and b(n) is pair of complements of natural numbers (A000027) with a(1) < a(2) < a(3) < ... and b(1) < b(2) < b(3) < ..., then F[a(n); b(n)] = F[a(n)] + F[b(n)]; where F[a(n)] = 1/a(1) + 1/a(1)a(2) + 1/a(1)a(2)a(3) + ... and F[b(n)] = 1/b(1) + 1/b(1)b(2) + 1/b(1)b(2)b(3) + ...
Value of function F[a(n); b(n)] is real number c = a + b, where a = real number whose Engel expansion is sequence a(n) and b = real number whose Engel expansion is sequence b(n). See A006784 for definition of Engel expansion.
Example for a(n) = odd numbers (A005408) and b(n) = even numbers (A005843): c = 2.059407... = a + b, where a = 1.410686... (A060196) and b = 0.648721... (A019774 - 1).
Example for a(n) = nonprime numbers (A018252) and b(n) = primes (A000040): c = 2.002747... = a + b, where a = 1.297516... and b = 0.705230... (A064648).
Conjecture: there are no pairs of complements a(n) and b(n) such that F[a(n); b(n)] = 2.
e - 1 <= F[a(n); b(n)] <= sqrt(e) + sqrt((e*Pi)/2)*erf(1/sqrt(2)) - 1.
1.71828182... (A091131) <= F[a(n); b(n)] <= 2.05940740....

			2.05940740534257614453947549923327861297725472633534020929971877980544281968...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Engel Expansion
Googology Wiki, Double factorial

Cf. A000027, A005408, A005843, A091131 (e-1), A006882 (n!!), A143280 (m(2)).

SetDefaultRealField(RealField(112)); R:= RealField(); -1 + Exp(1/2)*(1 + Sqrt(Pi(R)/2)*Erf(1/Sqrt(2)) ); // G. C. Greubel, Apr 01 2019

RealDigits[Sqrt[E] -1 + Sqrt[E*Pi/2]*Erf[1/Sqrt[2]], 10, 105][[1]] (* or *)
RealDigits[Sum[1/n!!, {n, 125}], 10, 105][[1]] (* Robert G. Wilson v, Apr 09 2014 *)

default(realprecision, 100); exp(1/2) - 1 + sqrt(exp(1)*Pi/2)*(1-erfc(1/sqrt(2))) \\ G. C. Greubel, Apr 01 2019

numerical_approx(-1 + exp(1/2)*(1 + sqrt(pi/2)*erf(1/sqrt(2))), digits=112) # G. C. Greubel, Apr 01 2019

A271476 Total number of burnt pancakes flipped using the Min-bar(n) greedy algorithm.

Original entry on oeis.org

1, 10, 75, 628, 6325, 75966, 1063615, 17017960, 306323433, 6126468850, 134782314931, 3234775558620, 84104164524445, 2354916606684838, 70647498200545575, 2260719942417458896, 76864478042193603025, 2767121209518969709530, 105150605961720848962843, 4206024238468833958514500

Offset: 1

N. J. A. Sloane, Apr 09 2016

nonn

Gheorghe Coserea, Table of n, a(n) for n = 1..128
J. Sawada, A. Williams, Successor rules for flipping pancakes and burnt pancakes, Preprint, Theoretical Computer Science, Volume 609, Part 1, 4 January 2016, Pages 60-75.

Cf. A019774, A093302.

List([1..20],n->-n+2^n*Factorial(n)*Sum([0..n-1],k->1/(2^k*Factorial(k)))); # Muniru A Asiru, Aug 02 2018

seq(coeff(series(factorial(n)*exp(x)*(x+2*x^2)/(1-2*x), x,n+1),x,n),n=1..20); # Muniru A Asiru, Aug 02 2018

Table[2^n*n! Sum[1/(2^k*k!), {k, 0, n - 1}] - n, {n, 20}] (* Michael De Vlieger, May 25 2016 *)

a(n) = 2^n * n! * sum(k=0, n-1, 1/(2^k*k!)) - n;
vector(20, n, a(n))  \\ Gheorghe Coserea, Apr 25 2016

x='x+O('x^99); Vec(serlaplace((x+2*x^2)/(1-2*x)*exp(x))) \\ Altug Alkan, Aug 01 2018

a(n) = -n + 2^n * n! * Sum_{k=0..n-1} 1/(2^k*k!). (see Sawada link) - Gheorghe Coserea, Apr 25 2016
From Altug Alkan, Aug 01 2018: (Start)
a(n) = A093302(n)/2 for n >= 1.
a(n) = floor(e^(1/2)*n!*2^n)-n-1.
E.g.f.: exp(x)*(x+2*x^2)/(1-2*x). (End)

More terms from Gheorghe Coserea, Apr 25 2016

A306486 Number of squares in the interval [e^(n-1), e^n).

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 13, 21, 36, 58, 96, 159, 262, 431, 712, 1172, 1934, 3189, 5256, 8667, 14289, 23559, 38841, 64039, 105583, 174076, 287003, 473188, 780155, 1286258, 2120681, 3496412, 5764609, 9504233, 15669832, 25835185, 42595018, 70227313, 115785266

Offset: 0

Alexei Kourbatov, Feb 18 2019

nonn

The lower endpoint e^(n-1) is included; the upper endpoint is not included. The terms a(0) to a(8) coincide with the Fibonacci numbers.

			Between exp(2) and exp(3) there are two squares, namely, 9 and 16; therefore, a(3)=2.

Alois P. Heinz, Table of n, a(n) for n = 0..4607 (first 501 terms from Alexei Kourbatov)

Cf. A000045, A000290, A001113, A005181, A019774, A290506, A306604.

a:= n-> (f-> f(n)-f(n-1))(i-> ceil(exp(i/2))):
seq(a(n), n=0..44);  # Alois P. Heinz, Feb 18 2019

a(n)=ceil(sqrt(exp(n)))-ceil(sqrt(exp(n-1)));
for(n=0,50,print1(a(n)", "))

a(n) = ceiling(sqrt(exp(n))) - ceiling(sqrt(exp(n-1))).
From Alois P. Heinz, Feb 19 2019: (Start)
Lim_{n->oo} a(n+1)/a(n) = sqrt(e) = 1.64872127... = A019774.
a(n) = A005181(n+1) - A005181(n). (End)
a(n) = (1-1/sqrt(e))*e^(n/2)+O(1) ~ 0.39346934...*e^(n/2) ~ A290506*e^(n/2). - Alexei Kourbatov, Feb 20 2019

A019784 Decimal expansion of sqrt(e)/11.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

			0.149883751881829831531695526164923961...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index entries for transcendental numbers.

Cf. A019774.

RealDigits[Sqrt[E]/11, 10, 100][[1]] (* Alonso del Arte, Feb 11 2018 *)

exp(.5)/11 \\ Charles R Greathouse IV, Nov 26 2024

A019790 Decimal expansion of sqrt(e)/17.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

			0.096983604158831067461685340459656680685516241218244000680887...

Ivan Panchenko, Table of n, a(n) for n = 0..1001

Cf. A019774.

RealDigits[Sqrt[E]/17,10,120][[1]] (* Harvey P. Dale, Sep 07 2021 *)

exp(1/2)/17 \\ Michel Marcus, Aug 25 2022

Equals A019774/17.

a(0) = 0 prepended by Georg Fischer, Aug 25 2022

cp's OEIS Frontend

A334367 Decimal expansion of Sum_{k>=0} 1/(4*k+2)!!.

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A019778 Decimal expansion of sqrt(e)/5.

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A092618 Decimal expansion of e^(-1/5).

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A092750 Decimal expansion of e^(-1/7).

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A181180 Decimal expansion of exp(exp(-1/2)).

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A227569 Decimal expansion of maximal value of function F[a(n); b(n)] for pairs of complements a(n) and b(n) of natural numbers A000027, where a(n) = odd numbers (A005408) and b(n) = even numbers (A005843); see Comments for the definition of function F[a(n); b(n)].

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A271476 Total number of burnt pancakes flipped using the Min-bar(n) greedy algorithm.

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A306486 Number of squares in the interval [e^(n-1), e^n).

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