A019774 -id:A019774 - cp's OEIS frontend

A331501 Decimal expansion of exp(3/4).

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

Offset: 1

Washington Bomfim, Feb 27 2020

Considering graph evolutions (see the Flajolet link) with 2n vertices initially isolated, the probability of the occurrence of an acyclic graph at the critical point n in the uniform model, will be denoted by P(n). In the case of the permutation model, the respective probability will be denoted by Pp(n).

Pp(n) / P(n) ~ exp(3/4) since Pp(n) = A302112(n) / A331505(2n) = A302112(n) / C(C(2n,2), n), and P(n) = A302112(n) * n! * 2^n / (2n)^(2n), Pp(n) / P(n) = (2n)^(2n) / (C(C(2n,2), n) * n! * 2^n), and lim_{n->oo} Pp(n) / P(n) = exp(3/4).

			2.1170000166126746685453698198370956101344915847024...

Philippe Flajolet, Donald E. Knuth, and Boris Pittel, The first cycles in an evolving graph, Discrete Mathematics, Vol. 75, No. 1-3 (1989), pp. 167-215.
Leonard Giugiuc and Dan Stefan Marinescu, Problem 4257, Crux Mathematicorum, Vol. 43, No. 6 (2017), pp. 263 and 265; Solution to Problem 4257, ibid., Vol. 44, No. 6 (2018), pp. 268-270.
Index entries for transcendental numbers

Cf. A000178, A001113, A019774, A092042, A302112, A331500, A331502, A331505.

Maple
```
evalf(exp(3/4), 134);
```

RealDigits[Exp[3/4], 10, 100][[1]] (* Amiram Eldar, Apr 12 2022 *)

exp(3/4) \\ Charles R Greathouse IV, Nov 21 2024

Equals lim_{n->oo} Pp(n) / P(n) = lim_{n->oo} (2*n)^(2*n) / (binomial(binomial(2n,2), n) * n! * 2^n).

Equals lim_{n->oo} sqrt(n)/A000178(n)^(1/(n*(n+1))) (Giugiuc and Marinescu, 2017). - Amiram Eldar, Apr 12 2022

A379411 a(n) = n + floor(ns/r) + floor(nt/r), where r = e^(1/4), s = e^(1/2), t = e^(3/4).

Original entry on oeis.org

3, 7, 10, 15, 19, 22, 26, 31, 34, 38, 43, 46, 50, 54, 58, 62, 66, 70, 74, 77, 81, 86, 89, 93, 98, 101, 105, 109, 113, 117, 121, 125, 129, 133, 136, 141, 145, 148, 153, 156, 160, 164, 168, 172, 176, 180, 184, 188, 191, 196, 200, 203, 208, 212, 215, 219, 223

Offset: 1

Clark Kimberling, Jan 18 2025

nonn

This sequence and A379412 and A379413 partition the positive integers; see A184812 for a proof. For each k in A000027, write "a" if k=A379411(n) for some n, "b" if k=A379412(n) for some n, and "c" if k=A379413(n) for some n. Concatenating these letters for k = 1,2,3,... spells the following infinite word:

cbacbcabcacbcbacbcabcacbcabcbcacbacbcabcbcacbacbcabccabcbacbcabccabcbacbcacbacbcabcbcacbacbcabccbacbacbcabccabcbacbcacbcabcbacbcacbcabcabccbacb...

Cf. A184812, A379412, A379413.

Cf. A092042, A019774, A331501.

r = E^(1/4); s = E^(1/2); t = E^(3/4);
Table[n + Floor[n*s/r] + Floor[n*t/r], {n, 1, 120}]  (* A379411 *)
Table[n + Floor[n*r/s] + Floor[n*t/s], {n, 1, 120}]  (* A379412 *)
Table[n + Floor[n*r/t] + Floor[n*s/t], {n, 1, 120}]  (* A379413 *)

a(n) = n + floor(n*r) + floor(n*r^2), where r = e^(1/4).

A379413 a(n) = n + floor(nr/t) + floor(ns/t), where r = e^(1/4), s = e^(1/2), t = e^(3/4).

Original entry on oeis.org

1, 4, 6, 9, 11, 13, 16, 18, 21, 23, 25, 28, 30, 32, 35, 37, 40, 42, 44, 47, 49, 52, 53, 56, 59, 61, 64, 65, 68, 71, 73, 75, 78, 80, 83, 85, 87, 90, 92, 95, 96, 99, 102, 104, 107, 108, 111, 114, 116, 118, 120, 123, 126, 128, 130, 132, 135, 138, 139, 142, 144

Offset: 1

Clark Kimberling, Jan 18 2025

nonn

This sequence and A379411 and A379412 partition the positive integers; see A378142 for a proof.

Cf. A378142, A379411, A379412.

Cf. A092042, A019774, A331501.

r = E^(1/4); s = E^(1/2); t = E^(3/4);
Table[n + Floor[n*s/r] + Floor[n*t/r], {n, 1, 120}]  (* A379411 *)
Table[n + Floor[n*r/s] + Floor[n*t/s], {n, 1, 120}]  (* A379412 *)
Table[n + Floor[n*r/t] + Floor[n*s/t], {n, 1, 120}]  (* A379413 *)

a(n) = n + floor(n/r) + floor(n*r^2), where r = e^(1/4).

A092515 Decimal expansion of e^(1/6).

Original entry on oeis.org

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

Offset: 1

Mohammad K. Azarian, Apr 05 2004

e^(1/6) maximizes the value of x^(c/(x^6)) for any real positive constant c, and minimizes for it for a negative constant, on the range x > 0. - A.H.M. Smeets, Aug 16 2018

			1.18136041286564598030511215250718432783018931083896...

Muniru A Asiru, Table of n, a(n) for n = 1..2000

Cf. A001113, A019774, A092727 (reciprocal).

evalf(exp(1/6)); # Muniru A Asiru, Aug 16 2018

RealDigits[E^(1/6),10,120][[1]] (* Harvey P. Dale, Sep 13 2011 *)

PARI
```
exp(1/6) \\ Michel Marcus, Aug 16 2018
```

Equals lim_{x->0} (sinh(x)/x)^(1/x^2). - Amiram Eldar, Jul 04 2022

A092554 Decimal expansion of e^(-3).

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Apr 09 2004

			0.049787068367863942979342415650061776631699592188423...

Cf. A019774, A001113, A068985.

First[RealDigits[E^-3, 10, 100, -1]] (* Paolo Xausa, Feb 09 2025 *)

PARI
```
exp(-3) \\ Michel Marcus, May 10 2022
```

A092555 Decimal expansion of e^(-4).

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Apr 09 2004

This is one of Rényi's parking constants. - Alonso del Arte, Dec 28 2013

			0.0183156388887...

Steven R. Finch, Mathematical Constants, Cambridge University Press (2003): 280.

Cf. A019774, A001113, A068985, A092553 (e^(-2)), A092554 (e^(-3)).

RealDigits[E^(-4), 10, 100][[1]] (* Alonso del Arte, Dec 28 2013 *)

A092616 Decimal expansion of e^(-1/4).

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Apr 22 2004

			0.778800783071404

Cf. A001113, A019774, A068985.

RealDigits[Surd[E,-4],10,120][[1]] (* Harvey P. Dale, Jul 07 2013 *)
Limit[Product[(k/n)^(k/n^2), {k, 1, n}], n->Infinity] (* Vaclav Kotesovec, Oct 06 2023 *)

Equals limit_{n->oo} Product_{k=1..n} (k/n)^(k/n^2). - Vaclav Kotesovec, Oct 06 2023

A092727 Decimal expansion of e^(-1/6).

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Apr 22 2004

			0.84648172489061407404491739979875457688829162442705...

Cf. A001113, A019774, A068985, A092515 (reciprocal).

RealDigits[E^-(1/6),10,120][[1]] (* Harvey P. Dale, Jun 18 2012 *)

Equals lim_{x->0} (sin(x)/x)^(1/x^2). - Amiram Eldar, Jul 04 2022

A143412 Main diagonal of A143410.

Original entry on oeis.org

1, 3, 37, 743, 20841, 751019, 33065677, 1720166223, 103243039057, 7022246822099, 533794001518581, 44845718374382903, 4126339884444745657, 412678834162848948603, 44573440429472131194781, 5170931768652930067543199, 641240112753392800506551457, 84648865815216502596932335523

Offset: 0

Peter Bala, Aug 14 2008

The sequence of convergents of the continued fraction expansion sqrt(e) = 1 + 2/(3 + 1/(12 + 1/(20 + 1/(28 + 1/(36 + ... ))))) begins [1/1, 5/3, 61/37, 1225/743, ...]. The partial denominators are this sequence; the numerators are A065919. - Peter Bala, Jan 02 2020

Robert Israel, Table of n, a(n) for n = 0..333

Cf. A065919, A143410, A019774.

I:=[1,3]; [n le 2 select I[n] else 4*(2*n -3)*Self(n - 1) + Self(n-2): n in [1..20]]; // Vincenzo Librandi, Jan 03 2016

a := n -> (-1)^n*add ((-2)^k*(n+k)!/((n-k)!*k!),k = 0..n): seq(a(n),n = 0..16);
seq(simplify(2^n*KummerU(-n,-2*n,-1/2)), n=0..17); # Peter Luschny, May 10 2022

RecurrenceTable[{ a[n + 2] == 4*(2 n + 3)*a[n + 1] + a[n], a[0] == 1, a[1] == 3}, a, {n, 0, 20}] (* G. C. Greubel, Jan 03 2016 *)

a(n) = (-1)^n*sum(k=0,n, (-2)^k*(n+k)!/((n-k)!*k!) ); \\ Joerg Arndt, May 17 2013

a(n) = (-1)^n*Sum_{k = 0..n} (-2)^k*(n+k)!/((n-k)!*k!) = (-1)^n*y_n(-4), where y_n(x) denotes the n-th Bessel polynomial.
Recurrence relation: a(0) = 1, a(1) = 3, a(n) = 4*(2*n-1)*a(n-1) + a(n-2) for n >= 2. Sequence A065919 satisfies the same recurrence relation.
sqrt(e) = 1 + 2*Sum_{n >= 0} (-1)^n/(a(n)*a(n+1)) = 1 + 2*(1/(1*3) - 1/(3*37) + 1/(37*743) - ...) (see A019774).
G.f.: 1/Q(0), where Q(k)= 1 + x - 4*x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 17 2013
a(n) = (-1)^n * hypergeom([-n,n+1],[],2). - Robert Israel, Jan 03 2016
a(n) ~ 2^(3*n + 1/2) * n^n / exp(n + 1/4). - Vaclav Kotesovec, Jan 02 2020
a(n) is the expectation of U_{2n}(X) where X is a standard Gaussian random variable and U_n is the n-th Chebyshev polynomial of second kind (conjectured). - Benjamin Dadoun, Dec 16 2020
a(n) = 2^n*KummerU(-n, -2*n, -1/2). - Peter Luschny, May 10 2022

A226161 Least positive integer k such that 1 + 1/2 + ... + 1/k > n/2.

Original entry on oeis.org

1, 2, 3, 4, 7, 11, 19, 31, 51, 83, 137, 227, 373, 616, 1015, 1674, 2759, 4550, 7501, 12367, 20390, 33617, 55425, 91380, 150661, 248397, 409538, 675214, 1113239, 1835421, 3026097, 4989191, 8225785, 13562027, 22360003, 36865412, 60780790, 100210581, 165219316

Offset: 1

Clark Kimberling, May 29 2013

nonn

Conjecture: a(n+1)/a(n) converges to 1.64872...
The conjecture is correct, a(n+1)/a(n) ~ exp(1/2) (A019774). - Charles R Greathouse IV, Jun 03 2013
Conjecture: a(n) = round(exp(n/2-gamma)) for all n, where gamma is the Euler-Mascheroni constant (see A001620). - Jon E. Schoenfield, Jul 19 2015
The terms up to a(52) contained in the b-file have been obtained by working with quadruple-precision (128 bits) floating point numbers, hence there is a very small probability they are off by 1. - Giovanni Resta, Jul 21 2015
All terms in the b-file are correct. Moreover, the above conjecture that a(n) = round(exp(n/2-gamma)) has been verified for all n in 1..10000. - Jon E. Schoenfield, Jul 22 2015

			a(5) = 7 because 1 + 1/2 + ... + 1/6 < 5/2 < 1 + 1/2 + ... + 1/6 + 1/7.

Giovanni Resta, Table of n, a(n) for n = 1..52

Cf. A001620, A019774, A226160.

nn = 24; g = 1/2; f[n_] := 1/n; a[1] = 1; Do[s = 0; a[n] = NestWhile[# + 1 &, 1, ! (s += f[#]) > n*g &], {n, nn}]; Map[a, Range[nn]]

first(m)=my(v=vector(m),i,k);for(i=1,m,k=1;while(sum(x=1,k,1/x)<=i/2,k++);v[i]=k;);v; \\ Anders Hellström, Jul 19 2015

a(29)-a(35) from Jean-François Alcover, Jun 04 2013
a(36)-a(37) from Jon E. Schoenfield, Aug 31 2013
a(38)-a(39) from Jon E. Schoenfield, Jul 19 2015

cp's OEIS Frontend

A331501 Decimal expansion of exp(3/4).

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A379411 a(n) = n + floor(n*s/r) + floor(n*t/r), where r = e^(1/4), s = e^(1/2), t = e^(3/4).

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A379413 a(n) = n + floor(n*r/t) + floor(n*s/t), where r = e^(1/4), s = e^(1/2), t = e^(3/4).

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A092515 Decimal expansion of e^(1/6).

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A092554 Decimal expansion of e^(-3).

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A092555 Decimal expansion of e^(-4).

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A092616 Decimal expansion of e^(-1/4).

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A092727 Decimal expansion of e^(-1/6).

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A379411 a(n) = n + floor(ns/r) + floor(nt/r), where r = e^(1/4), s = e^(1/2), t = e^(3/4).

A379413 a(n) = n + floor(nr/t) + floor(ns/t), where r = e^(1/4), s = e^(1/2), t = e^(3/4).