A068985 -id:A068985 - cp's OEIS frontend

A135004 Decimal expansion of 4/e.

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

Offset: 1

Omar E. Pol, Nov 15 2007

Leading coefficient of energy difference between the two lowest states of the hydrogen molecular ion (see Holstein-Herring Method link). - Andrea Pinos, Nov 22 2023

			1.471517764685769...

G. C. Greubel, Table of n, a(n) for n = 1..2000
Shawn Godin, Problem CC35, Crux Mathematicorum, Vol. 38, No. 7 (2012), p. 265; Solution to Problem CC35, ibid., Vol. 39, No. 7 (2013), p. 299.
Wikipedia, Holstein-Herring Method.

Cf. A019741 (e/4), A068985 (1/e), A135002 (2/e).

Digits:=100: evalf(4/exp(1)); # Wesley Ivan Hurt, Jan 19 2017

RealDigits[4/E,10,120][[1]] (* Harvey P. Dale, Mar 06 2012 *)

PARI
```
4/exp(1) \\ Michel Marcus, Jan 20 2017
```

Equals lim_{n->oo} (((2*n)!/n!)^(1/n))/n (Godin, 2012). - Amiram Eldar, Apr 02 2022

More terms from Harvey P. Dale, Mar 06 2012

A135005 Decimal expansion of 5/e.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Nov 15 2007

The fraction of substituents which become isolated in a simple polymer model is 1/10 this number, see Flory 1939 (and 1936). - Charles R Greathouse IV, Nov 30 2012

			5/e = 1.839397205857211607977618850807304337229...
1/(2*e) = 0.1839397205857211607977618850807304337229... (see Greathouse's comment).

G. C. Greubel, Table of n, a(n) for n = 1..2000
Paul J. Flory, Intramolecular reaction between neighboring substituents of vinyl polymers, Journal of the American Chemical Society 61:6 (1939), pp. 1518-1521.
Paul J. Flory, Molecular size distribution in linear condensation polymers, Journal of the American Chemical Society 58:10 (1936), pp. 1877-1885.
Ovidiu Furdui, From Lalescu's sequence to a Gamma function limit, Gazette of the Australian Mathematical Society, Vol. 35, No. 5 (2008), pp. 339-344.

Cf. A001113, A068985.

RealDigits[5/E , 10, 50][[1]] (* G. C. Greubel, Sep 16 2016 *)

PARI
```
5/exp(1) \\ Michel Marcus, Sep 17 2016
```

Equals 5/A001113 and 5*A068985. - Michel Marcus, Sep 17 2016
1/(2*e) = Integral_{x=1..oo} e^(-x^2) * x dx. - Amiram Eldar, Aug 03 2020
1/(2*e) = lim_{n->oo} n * ((n+1)!^(1/(n+1)) - n!^(1/n) - 1/e) (Furdui, 2008). - Amiram Eldar, Apr 20 2023

A181590 Least value of n such that |P(n) - 1/e| < 10^(-i), i=1,2,3... . P(n)=floor(n!/e + 1/2)/n! is the probability of a random permutation on n objects be a derangement.

Original entry on oeis.org

3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 14, 15, 16, 17, 18, 18, 19, 20, 21, 21, 22, 23, 24, 24, 25, 26, 26, 27, 28, 29, 29, 30, 31, 31, 32, 33, 33, 34, 34, 35, 36, 36, 37, 38, 38, 39, 40, 40, 41, 41, 42, 43, 43, 44, 44, 45, 46, 46, 47, 47, 48, 49, 49, 50, 50, 51, 52, 52, 53, 53

Offset: 1

Washington Bomfim, Oct 31 2010

nonn

Both P(n) and the probability that a rooted forest on [n] be a tree tend to 1/e when n rises to infinity. So the events random forest be a tree and random permutation be a derangement become equiprobable as n tends to infinity.

The probability P(n) approaches 1/e quite quickly as this sequence shows. See image clicking the first link.

			a(2) = 4, a(3) = 6, so for n in the interval 4...5, if we use 1/e as the probability P, we make an error less than 10^(-1).
In general if n is in the interval a(i), ... , a(i+k)-1, k the least positive integer such that a(i+k) > a(i), this error is less than 10(-i-k+1).
For example, a(11) = a(12) = 14, k = 2 and if n is in the interval 14...14, if we use 1/e as the probability P, we make an error less than 10^(-12).

ttmath, C++ multiprecision math software
Wikipedia, Graph of probabilities
Wikipedia, The number e

Cf. A068985, A000166, A181589.

$MaxExtraPrecision = 100; f[n_] := Block[{k = 1}, While[ Abs[ Floor[(k!/E + 1/2)]/k! - 1/E] > 1/10^n, k++ ]; k]; Array[f, 71] (* Robert G. Wilson v, Nov 05 2010 *)

More terms from Robert G. Wilson v, Nov 05 2010

A182767 Beatty sequence for 1+e^2.

Original entry on oeis.org

8, 16, 25, 33, 41, 50, 58, 67, 75, 83, 92, 100, 109, 117, 125, 134, 142, 151, 159, 167, 176, 184, 192, 201, 209, 218, 226, 234, 243, 251, 260, 268, 276, 285, 293, 302, 310, 318, 327, 335, 343, 352, 360, 369, 377, 385, 394, 402, 411, 419, 427

Offset: 1

Clark Kimberling, Nov 29 2010

nonn

Let u=e=A001113 and v=1/e=A068985. Jointly rank {j*u} and {k*v} as in the first comment at A182760; a(n) is the position of n*u.

Cf. A182760, A182768.

A182767 := proc(n) floor(n*(1+exp(2))) ; end proc:

a(n)=floor(n*(1+e^2)) = floor(n+n*A072334).

A278327 Decimal expansion of 1/e - 1/e^2.

Original entry on oeis.org

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

Offset: 0

Alois P. Heinz, Nov 18 2016

The probability, as n = 2^k increases without bound, that a randomized skip list with n elements and p = 1/2 has exactly k levels.

			0.232544157934829629701524275188976464...

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Wikipedia, Skip list.

Cf. A001113, A068985, A072334, A091131, A092553, A158466.

RealDigits[1/E - 1/E^2, 10, 120][[1]] (* Amiram Eldar, May 27 2023 *)

Equals 1/A001113 - 1/A072334 = A068985 - A092553.

Equals A091131 / A072334 = A091131 * A092553.

A282822 a(n) = (n - 4)*n! for n>=0.

Original entry on oeis.org

-4, -3, -4, -6, 0, 120, 1440, 15120, 161280, 1814400, 21772800, 279417600, 3832012800, 56043187200, 871782912000, 14384418048000, 251073478656000, 4623936565248000, 89633231880192000, 1824676506132480000, 38926432130826240000, 868546016919060480000

Offset: 0

Bruno Berselli, Feb 22 2017

Cf. A034865.
Cf. sequences with formula (n + k)*n! listed in A282466.
Cf. A001113, A001620, A068985, A091725, A099285.

Table[(n - 4) n!, {n, 0, 30}] (* or *)
RecurrenceTable[{a[0] == -4, a[n] == n a[n - 1] + n!}, a, {n, 0, 30}]

E.g.f.: -(4 - 5*x)/(1 - x)^2.
a(n) = n*a(n-1) + n!, with n>0, a(0)=-4.
a(n) = 2*A034865(n) for n>3.
From Amiram Eldar, Jan 14 2021: (Start)
Sum_{n>=5} 1/a(n) = 313/288 - 5*e/12 - gamma/24 + Ei(1)/24 = 313/288 - (5/12)*A001113 - (1/24)*A001620 + A091725/24.
Sum_{n>=5} (-1)^(n+1)/a(n) = -25/288 + 1/(6*e) + gamma/24 - Ei(-1)/24 = -25/288 - (1/6)*A068985 + (1/24)*A001620 + (1/24)*A099285. (End)

A325748 First term of n-th difference sequence of (floor(k/e)), k >= 0.

Original entry on oeis.org

0, 0, 1, -3, 6, -9, 9, 0, -27, 81, -161, 231, -165, -363, 2079, -6435, 16071, -35343, 70686, -130339, 222717, -352715, 515944, -692967, 849299, -950455, 996435, -1042415, 1042415, 1, -6926071, 36516481, -137549665, 436362817, -1233496132, 3197432557

Offset: 1

Clark Kimberling, Jun 07 2019

Clark Kimberling, Table of n, a(n) for n = 1..200

Cf. A068985.

Table[First[Differences[Table[Floor[n/E], {n, 0, 50}], n]], {n, 1, 50}]

A337986 Prime numbers p such that v_p(A000166(k)) = v_p(k-1) for all k > 1, where v_p(k) is the p-adic valuation of k.

Original entry on oeis.org

2, 5, 7, 17, 19, 23, 29, 43, 59, 61, 71, 73, 107, 113, 131, 137, 149, 157, 173, 181, 191, 197, 199, 229, 233, 239, 241, 251, 257, 311, 317, 331, 349, 383, 401, 409, 421, 461, 491, 499, 541, 547, 557, 599, 601, 613, 619, 641, 653, 673, 719, 751, 761, 787, 797, 809

Offset: 1

Amiram Eldar, Jan 29 2021

nonn

Miska (2016) proved that the complement of this sequence within the primes is infinite, and conjectured that this sequence is also infinite, and that its asymptotic density within the primes is 1/e (A068985). Numerically, he found that there are 28990 terms below 10^6, which are about 37% of all the primes less than 10^6.

			2 is a term since A007814(A000166(k)) = A007814(k-1) for all k > 1.

Amiram Eldar, Table of n, a(n) for n = 1..1000
Piotr Miska, Arithmetic properties of the sequence of derangements, Journal of Number Theory, Vol. 163 (2016), pp. 114-145; arXiv preprint, arXiv:1508.01987 [math.NT], 2015.

Cf. A000166, A000255, A007814, A068985.

e[n_] := e[n] = Subfactorial[n]/(n - 1); q[p_] := PrimeQ[p] && AllTrue[Table[e[n], {n, 2, p + 1}], ! Divisible[#, p] &]; Select[Range[1000], q]

A prime p is a term if and only if p does not divide any of the numbers A000255(k), k in {2, ..., p+1}.

A365594 The denominators of a series that converges to 1/e obtained using Whittaker's Root Series Formula.

Original entry on oeis.org

3, 42, 154, 3817, 1141283, 119706444, 1396550916, 20958700652, 2359646218028, 324742403298918, 107268957934572210, 41877140987048387615, 19073758392921536694655, 10024177256513161424322680, 376301673554116445531842536, 10673126660749797308728534491

Offset: 1

Raul Prisacariu, Sep 10 2023

The Whittaker's Root Series Formula is applied to 1+x-x^2/2+x^3/3-x^4/4+x^5/5-x^6/6 +..., which is 1 + the Taylor expansion of log(1+x). The series obtained after applying Whittaker's Root Series Formula: 1/e-1=(-1)/1+(1/2)/(1*3/2)+(1/12)/((3/2)*(7/3))+(1/18)/((7/3)*(11/3))+(1/20)/((11/3)*(347/60))+(563/10800)/((347/60)*(3289/360))+ ... . The series can be simplified to: 1/e=1/3+1/42+1/154+9/3817+1126/1141283+ ... . The sequence is formed by the denominators of the simplified series.

The fractions in the denominators of the non-simplified series seem to be equal to terms from A323339 divided by the corresponding terms from A323340. Thus, the Whittaker's Root Series for 1 + the Taylor expansion of log(1+x) offers an alternative method for obtaining the terms of A323339 and A323340 using the determinants of Toeplitz matrices (formed using the coefficients of 1 + the Taylor expansion of log(1+x)).

			Whittaker's Root Series Formula is applied to 1 + the Taylor expansion of log(1+x) and the terms are simplified. The sequence is formed by the denominators of the simplified terms, starting with the second term in the Whittaker's Root Series.
a(1) is the denominator of -(-1/2)/(1*det((1,-1/2),(1,1))) = (1/2)/(3/2) = 1/3.
a(2) is the denominator of -det((-1/2,1/3),(1,-1/2))/(det((1,-1/2),(1,1))*det((1,-1/2,1/3),(1,1,-1/2),(0,1,1))) = (1/12)/((3/2)*(7/3)) = 1/42.

Raul Prisacariu, Whittaker's Root Series: Going Transcendental.
E. T. Whittaker and G. Robinson, The Calculus of Observations, London: Blackie & Son, Ltd. 1924, pp. 120-123.

Cf. A068985 (1/e), A365595 (numerators), A323339, A323340.

c[k_] := If[k < 0, 0, SeriesCoefficient[1 + Log[1 + x], {x, 0, k}]]; Table[-Det[ToeplitzMatrix[Table[c[3 - j], {j, 1, n}], Table[c[j + 1], {j, 1, n}]]] / (Det[ToeplitzMatrix[Table[c[2 - j], {j, 1, n}], Table[c[j], {j, 1, n}]]] * Det[ToeplitzMatrix[Table[c[2 - j], {j, 1, n + 1}], Table[c[j], {j, 1, n + 1}]]]), {n, 1, 20}] // Denominator (* Vaclav Kotesovec, Oct 09 2023 *)

a(n) is the denominator of the simplified fraction -det ToeplitzMatrix((c(2),c(1),c(0),0,0,...,0),(c(2),c(3),c(4),...,c(n+1)))/(det ToeplitzMatrix((c(1),c(0),0,...,0),(c(1),c(2),c(3),...,c(n)))*det ToeplitzMatrix((c(1),c(0),0,...,0),(c(1),c(2),c(3),...,c(n+1)))), where c(0)=1, c(1)=1, c(2)=-1/2, c(3)=1/3, c(4)=-1/4, c(n)=(1/n)*(-1)^(n+1).c(n) is simply the coefficient of x^n in the series formed by 1+ the Taylor expansion of log(1+x).

a(6)-a(7) corrected and extended by Vaclav Kotesovec, Oct 09 2023

A365595 The numerators of a series that converges to 1/e obtained using Whittaker's Root Series Formula.

Original entry on oeis.org

1, 1, 1, 9, 1126, 53825, 302989, 2285199, 133296721, 9731109349, 1737376806937, 372236638394027, 94229801087550639, 27818002500902930641, 591930814558449521261, 9591188150350759241842, 2816408483135723327055984, 1394771058490469072473603553, 385768133102988434073147277769

Offset: 1

Raul Prisacariu, Sep 10 2023

The Whittaker's Root Series Formula is applied to 1+x-x^2/2+x^3/3-x^4/4+x^5/5-x^6/6 +..., which is 1 + the Taylor expansion of log(1+x). The series obtained after applying Whittaker's Root Series Formula: 1/e-1=(-1)/1+(1/2)/(1*3/2)+(1/12)/((3/2)*(7/3))+(1/18)/((7/3)*(11/3))+(1/20)/((11/3)*(347/60))+(563/10800)/((347/60)*(3289/360))+ ... . The series can be simplified to: 1/e=1/3+1/42+1/154+9/3817+1126/1141283+ ... . The sequence is formed by the numerators of the simplified series.

The fractions in the denominators of the non-simplified series seem to be equal to terms from A323339 divided by the corresponding terms from A323340. Thus, the Whittaker's Root Series for 1 + the Taylor expansion of log(1+x) offers an alternative method for obtaining the terms of A323339 and A323340 using the determinants of Toeplitz matrices (formed using the coefficients of 1 + the Taylor expansion of log(1+x)).

			Whittaker's Root Series Formula is applied to 1 + the Taylor expansion of log(1+x) and the terms are simplified. The sequence is formed by the numerators of the simplified terms, starting with the second term in the Whittaker's Root Series.
a(1) is the numerator of -(-1/2)/(1*det((1,-1/2),(1,1))) = (1/2)/(3/2) = 1/3.
a(2) is the numerator of -det((-1/2,1/3),(1,-1/2))/(det((1,-1/2),(1,1))*det((1,-1/2,1/3),(1,1,-1/2),(0,1,1))) = (1/12)/((3/2)*(7/3)) = 1/42.

Raul Prisacariu, Whittaker's Root Series: Going Transcendental.
E. T. Whittaker and G. Robinson, The Calculus of Observations, London: Blackie & Son, Ltd. 1924, pp. 120-123.

Cf. A068985 (1/e), A365594 (denominators), A323339, A323340.

c[k_] := If[k < 0, 0, SeriesCoefficient[1 + Log[1 + x], {x, 0, k}]]; Table[-Det[ToeplitzMatrix[Table[c[3 - j], {j, 1, n}], Table[c[j + 1], {j, 1, n}]]] / (Det[ToeplitzMatrix[Table[c[2 - j], {j, 1, n}], Table[c[j], {j, 1, n}]]] * Det[ToeplitzMatrix[Table[c[2 - j], {j, 1, n + 1}], Table[c[j], {j, 1, n + 1}]]]), {n, 1, 20}] // Numerator (* Vaclav Kotesovec, Oct 09 2023 *)

a(n) is the numerator of the simplified fraction -det ToeplitzMatrix((c(2),c(1),c(0),0,0,...,0),(c(2),c(3),c(4),...,c(n+1)))/(det ToeplitzMatrix((c(1),c(0),0,...,0),(c(1),c(2),c(3),...,c(n)))*det ToeplitzMatrix((c(1),c(0),0,...,0),(c(1),c(2),c(3),...,c(n+1)))), where c(0)=1, c(1)=1, c(2)=-1/2, c(3)=1/3, c(4)=-1/4, c(n)=(1/n)*(-1)^(n+1).c(n) is simply the coefficient of x^n in the series formed by 1+ the Taylor expansion of log(1+x).

More terms from Vaclav Kotesovec, Oct 09 2023

cp's OEIS Frontend

A135004 Decimal expansion of 4/e.

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A135005 Decimal expansion of 5/e.

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A181590 Least value of n such that |P(n) - 1/e| < 10^(-i), i=1,2,3... . P(n)=floor(n!/e + 1/2)/n! is the probability of a random permutation on n objects be a derangement.

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A182767 Beatty sequence for 1+e^2.

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A278327 Decimal expansion of 1/e - 1/e^2.

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A282822 a(n) = (n - 4)*n! for n>=0.

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A325748 First term of n-th difference sequence of (floor(k/e)), k >= 0.

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A337986 Prime numbers p such that v_p(A000166(k)) = v_p(k-1) for all k > 1, where v_p(k) is the p-adic valuation of k.

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