A185393 -id:A185393 - cp's OEIS frontend

A063048 Numbers n such that the Reverse and Add! trajectory of n (presumably) does not reach a palindrome and does not join the trajectory of any term m < n.

196, 879, 1997, 7059, 10553, 10563, 10577, 10583, 10585, 10638, 10663, 10668, 10697, 10715, 10728, 10735, 10746, 10748, 10783, 10785, 10787, 10788, 10877, 10883, 10963, 10965, 10969, 10977, 10983, 10985, 12797, 12898, 13097, 13197, 13694

Offset: 1

Klaus Brockhaus, Jul 07 2001, revised Nov 04 2003

The starting number n is regarded as part of the trajectory, so palindromes are excluded from the sequence. A088753 is obtained if palindromes are not excluded. The smallest term in A063048 but not in A088753 is 19098, the smallest term in A088753 but not in A063048 is 9999.
Subsequence of A023108. Sequence A070788 is similarly defined, but palindromes are irrelevant. Corresponding sequences for other bases are A075252 (base 2), A077405 (base 3), A075421 (base 4).
If the trajectory of a number k joins the trajectory of a smaller number which is a term of the present sequence, then this occurs after very few Reverse and Add! steps (at most 8 for k < 100000, at most 10 for k < 1000000). On the other hand, the trajectories of the terms < 14000 do not join the trajectory of any smaller term within at least 1500 steps. This is the precise meaning of "presumably" in the definition.
The terms are rather unevenly distributed. They form clusters, especially above 10^4, 10^5, 10^6, ... . The interval from 10000 to 11000 for example contains 26 terms, whereas only two terms occur in the interval from 90000 to 100000.
It seems that if the most significant digit is not equal to 1, the least significant digit is always 9, while this does not hold for the Lychrel numbers as in A023108. - A.H.M. Smeets, Feb 18 2019
From A.H.M. Smeets, Sep 18 2021: (Start)
Let d_0 d_1 d_2 ... d_n be the decimal digits of an (n+1)-digit number.
All numbers in this sequence seem to satisfy the following condition:
d_0 = "1" or d_n = "9", and for all k, 0 < k < floor((n-1)/2), d_k = "0" or d_k = "9" or d_(n-k) = "0" or d_(n-k) = "9".
The plot log_10(a(n)) versus log_10(n) shows a stepwise behavior. However, the global behavior seems to be a straight line with slope e/(e-1) (= A185393). This slope is also obtained for the seeds in the Reverse and Add! problem in other bases. (End)

			1997 is a term since the trajectory of 1997 (presumably) does not lead to a number which occurs in the trajectory of 196 or of 879 (actually checked for the first 10000 terms of these trajectories). The trajectory of 1997 joins the trajectory of 106 at 97768 (cf. A070796), but 106 is not a term of the present sequence.

Daniel Lignon, Dictionnaire de (presque) tous les nombres entiers, Ellipses, Paris, 2012, 702 pages. See Entry 196.

A.H.M. Smeets, Table of n, a(n) for n = 1..21491 (terms 1..1711 from Klaus Brockhaus).
Klaus Brockhaus, Illustration: Distribution of terms below 2000000
Klaus Brockhaus, List of terms below 2000000
A.H.M. Smeets, log_10(a(n)) versus log_10(n)
Index entries for sequences related to Reverse and Add!

Cf. A033865, A023108, A006960, A063049 - A063065, A088753, A070788, A075252, A077405, A075421, A070796.

limit = 10^3; (* Assumes that there is no palindrome if none is found before "limit" iterations *)
utraj = {};
Select[Range[0, 14000], (x = NestWhileList[ # + IntegerReverse[#] &, #, ! PalindromeQ[#] &, 1, limit];
   If[Length[x] >= limit && Intersection[x, utraj] == {},
    utraj = Union[utraj, x]; True,
utraj = Union[utraj, x]]) &] (* Robert Price, Oct 16 2019 *)

A068996 Decimal expansion of 1 - 1/e.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Apr 08 2002

From the "derangements" problem: this is the probability that if a large number of people are given their hats at random, at least one person gets their own hat.
1-1/e is the limit to which (1 - !n/n!) {= 1 - A000166(n)/A000142(n) = A002467(n)/A000142(n)} converges as n tends to infinity. - Lekraj Beedassy, Apr 14 2005
Also, this is lim_{n->inf} P(n), where P(n) is the probability that a random rooted forest on [n] is not connected. - Washington Bomfim, Nov 01 2010
Also equals the mode of a Gompertz distribution when the shape parameter is less than 1. - Jean-François Alcover, Apr 17 2013
The asymptotic density of numbers with an even number of trailing zeros in their factorial base representation (A232744). - Amiram Eldar, Feb 26 2021

			0.6321205588285576784044762...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.3, pp. 12-17.
Anders Hald, A History of Probability and Statistics and Their Applications before 1750, Wiley, NY, 1990 (Chapter 19).
John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 65.

Brian Conrey and Tom Davis, Derangements.
Brady Haran and Tom Crawford, A Problem with Infinite Hotel Keys, YouTube Numberphile video, 2025.
MathOverflow, What is the effect of adding 1/2 to a continued fraction?.
Jonathan Sondow and Eric Weisstein, e, MathWorld.
Bala Subramanian, Why time constant is 63.2% not a 50 or 70%? (2018).
Index entries for transcendental numbers

Cf. A000166, A068985, A091131, A185393, A232744.

evalf[140](1-exp(-1));  # Alois P. Heinz, May 16 2025

RealDigits[1 - 1/E, 10, 100][[1]] (* Alonso del Arte, Jul 06 2012 *)

1 - exp(-1) \\ Michel Marcus, Mar 04 2016

Equals Integral_{x = 0 .. 1} exp(-x) dx. - Alonso del Arte, Jul 06 2012
Equals -Sum_{k>=1} (-1)^k/k!. - Bruno Berselli, May 13 2013
Equals Sum_{k>=0} 1/((2*k+2)*(2*k)!). - Fred Daniel Kline, Mar 03 2016
From Peter Bala, Nov 27 2019: (Start)
1 - 1/e =  Sum_{n >= 0} n!/(A(n)*A(n+1)), where A(n) = A000522(n).
Continued fraction expansion: [0; 1, 1, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, ...].
Related continued fraction expansions include
2*(1 - 1/e) = [1; 3, 1, 3, 1, 1, 1, 3, 3, 3, 1, 3, 1, 3, 5, 3, 1, 5, ..., 1, 3, 2*n + 1, 3, 1, 2*n + 1, ...];
(1/2)*(1 - 1/e) = [0; 3, 6, 10, 14, 18, ..., 4*n + 2, ...];
4*(1 - 1/e) = [2; 1, 1, 8, 3, 1, 1, 1, 1, 7, 1, 1, 2, 1, 1, 1, 2, 7, 1, 2, 2, 1, 1, 1, 3, ..., 7, 1, n, 2, 1, 1, 1, n+1, ...];
(1/4)*(1 - 1/e) = [0; 6, 3, 20, 7, 36, 11, 52, 15, ..., 16*n + 4, 4*n + 3, ...]. (End)
Equals Integral_{x=0..1} x * cosh(x) dx. - Amiram Eldar, Aug 14 2020
Equals A091131/e. - Hugo Pfoertner, Aug 20 2024

A086787 a(n) = Sum_{i=1..n} ( Sum_{j=1..n} i^j ).

Original entry on oeis.org

1, 8, 56, 494, 5699, 82200, 1419760, 28501116, 651233661, 16676686696, 472883843992, 14705395791306, 497538872883727, 18193397941038736, 714950006521386976, 30046260016074301944, 1344648068888240941017

Offset: 1

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Aug 04 2003

nonn

p divides a(p+1) for all prime p except 3. p^2 divides a(p+1) for prime p in A123374.
2 divides a(n) for n = {2, 3, 4, 6, 7, 8, 10, 11, 12, 14, 15, 16, 18, 19, 20, 22, 23, 24, 26, 27, 28, 30, 31, 32, 34, 35, 36, 38, 39, 40, 42, 43, 44, 46, 47, 48, 50, ...}.
2^2 divides a(n) for n = {2, 3, 6, 7, 8, 10, 11, 14, 15, 16, 18, 19, 22, 23, 24, 26, 27, 30, 31, 32, 34, 35, 38, 39, 40, 42, 43, 46, 47, 48, 50, ...}.
2^3 divides a(n) for n = {2, 3, 6, 7, 10, 11, 14, 15, 16, 18, 19, 22, 23, 26, 27, 30, 31, 32, 34, 35, 38, 39, 42, 43, 46, 47, 48, 50, ...}.
2^4 divides a(n) for n = {7, 14, 15, 18, 23, 30, 31, 32, 34, 39, 46, 47, 50, ...}.
2^5 divides a(n) for n = {15, 30, 31, 34, 47, 62, 63, 64, 66, 79, 94, 95, 98, ...}.
2^6 divides a(n) for n = {31, 62, 63, 66, 95, ...}.
2^7 divides a(n) for n = {63, 126, 127, 130, ...}.
It appears that for k > 2 the least few n such that a(n) is divisible by 2^(k+1) are n = {(2^k-1), 2*(2^k-1), 2*(2^k-1)+1, 2*(2^k-1)+3, 3*(2^k-1)+2, 4*(2^k-1)+2, 4*(2^k-1)+3, 4*(2^k-1)+4, 4*(2^k-1)+6, 5*(2^k-1)+4, 6*(2^k-1)+4, 6*(2^k-1)+5, 6*(2^k-1)+8, ...}. - Alexander Adamchuk, Oct 08 2006
Numbers n that divide a(n) are listed in A014741. - Alexander Adamchuk, Nov 03 2006

			a(2) = 8 = 1 + 1 + 2 + 4 = 1^1 + 1^2 + 2^1 + 2^2.

G. C. Greubel, Table of n, a(n) for n = 1..385

Cf. A000295, A014741, A215084, A349117.

seq(1-Psi(n)-gamma+sum(i^(n+1)/(i-1),i = 2 .. n),n=1..20);

Table[Sum[i^j,{i,1,n},{j,1,n}],{n,1,24}] (* Alexander Adamchuk, Oct 08 2006 *)
Table[ n + Sum[ i*(i^n-1)/(i-1), {i,2,n} ], {n,1,17} ] (* Alexander Adamchuk, Nov 03 2006 *)

a(n)=sum(i=1,n,sum(j=1,n,i^j)) \\ Charles R Greathouse IV, Jul 19 2013

a(n)=round(1-psi(n)-Euler+sum(i=2,n,i^(n+1)/(i-1))) \\ Charles R Greathouse IV, Jul 19 2013

def A086787(n): return sum(i**j for i in range(1,n+1) for j in range(1,n+1)) # Chai Wah Wu, Jan 08 2022

from sympy import digamma, EulerGamma
from fractions import Fraction
def A086787(n): return 1-digamma(n)-EulerGamma + sum(Fraction(i**(n+1),i-1) for i in range(2,n+1)) # Chai Wah Wu, Jan 08 2022

1 - Psi(n) - gamma + Sum_{i=2..n} (i^(n+1)/(i-1)), where Psi(n) is the digamma function and gamma is Euler's constant.
a(n) = Sum[ i^j, {i,1,n}, {j,1,n} ] = n + Sum[ i*(i^n - 1)/(i - 1), {i,2,n} ]. - Alexander Adamchuk, Nov 03 2006
a(n) = Sum_{k=1..n} (B(k+1, n+1) - B(k+1, 1))/(k+1), where B(n, x) are the Bernoulli polynomials. - Daniel Suteu, Jun 25 2018
a(n) ~ c * n^n, where c = 1 / (1 - exp(-1)) = A185393 = 1.58197670686932642438... - Vaclav Kotesovec, Nov 06 2021

Edited by Max Alekseyev, Jan 29 2012

A073333 Decimal expansion of 1/(e - 1) = Sum_{k >= 1} exp(-k).

Original entry on oeis.org

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

Offset: 0

Robert G. Wilson v, Aug 22 2002

The value of the general continued fraction with the partial numerators (A000027) and the partial denominators (A000027). The value of the fractional limit of the numerators (A000166) and the denominators (A002467). Abs(A002467/(e-1)-A000166)->0. - Seiichi Kirikami, Oct 30 2011

			0.581976706869326424385002005109011558546869301075396136266787059648...

Calvin C. Clawson, Mathematical Mysteries: The Beauty and Magic of Numbers, Springer, 2013. See p. 225.
Wolfram Research, Mathematica, Version 4.1.0.0, Help Browser, under the function NSumExtraTerms

G. C. Greubel, Table of n, a(n) for n = 0..20000
Mohammad K. Azarian, A Limit Expression of 1/(e-1), Problem # 799, College Mathematics Journal, Vol. 36, No. 2, March 2005, p. 161. Solution appeared in Vol. 37, No. 2, March 2006, pp. 147-148.
Mohammad K. Azarian, Euler's Number Via Difference Equations, International Journal of Contemporary Mathematical Sciences, Vol. 7, 2012, No. 22, pp. 1095-1102.
H. W. Gould, A rearrangement of series based on a partition of the natural numbers, The Fibonacci Quarterly, Vol. 15, No. 1 (1977), pp. 67-72.
Don Redmond, The Evaluation of Integral_{x=0..1} floor(-ln(x)) dx, Problem #153, Advanced Problem Archive, Missouri State University.
Michel Waldschmidt, Continued fractions, Ecole de recherche CIMPA-Oujda, Théorie des Nombres et ses Applications, 18-29 mai 2015: Oujda (Maroc).
Eric Weisstein's World of Mathematics, Continued Fraction Constants.
Eric Weisstein's World of Mathematics, Generalized Continued Fraction.
Index entries for transcendental numbers.

Cf. A001113, A000027, A000166, A002467, A185393, A194807, A307178.

1/(Exp(1) - 1); // G. C. Greubel, Apr 09 2018

h:=x->sum(1/exp(n),n=1..x); evalf[110](h(1500)); evalf[110](h(4000));

RealDigits[N[Sum[Exp[-n], {n, 1, Infinity}], 120]][[1]]
RealDigits[1/(E - 1), 10, 120][[1]] (* Eric W. Weisstein, May 08 2013 *)

suminf(k=1,exp(-k)) \\ Charles R Greathouse IV, Oct 04 2011

1/(exp(1)-1) \\ Charles R Greathouse IV, Oct 04 2011

Equals 1/(exp(1)-1). - Joseph Biberstine (jrbibers(AT)indiana.edu), Nov 03 2004
Also the unique real solution to log(1+x) - log(x) = 1. Equals 1-1/(1+1/(exp(1)-2)). Continued fraction is [0:1, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, ...]. - Gerald McGarvey, Aug 14 2004
Equals Sum_{n>=0} B_n/n!, where B_n is a Bernoulli number. - Fredrik Johansson, Oct 18 2006
1/(e-1) = 1/(1+2/(2+3/(3+4/(4+5/(5+...(continued fraction)))))). - Philippe Deléham, Mar 09 2013
Equals Integral_{x=0..oo} floor(x)*exp(-x). - Jean-François Alcover, Mar 20 2013
From Peter Bala, Oct 09 2013: (Start)
Equals (1/2)*Sum_{n >= 0} 1/sinh(2^n). (Gould, equation 22).
Define s(n) = Sum_{k = 1..n} 1/k! for n >= 1. Then 1/(e - 1) = 1 - Sum_{n >= 1} 1/( (n+1)!*s(n)*s(n+1) ) is a rapidly converging series of rationals (see A194807). Equivalently, 1/(e - 1) = 1 - 1!/(1*3) - 2!/(3*10) - 3!/(10*41) - ..., where [1, 3, 10, 41, ... ] is A002627.
We also have the alternating series 1/(e - 1) = 1!/(1*1) - 2!/(1*4) + 3!/(4*15) - 4!/(15*76) + ..., where [1, 1, 4, 15, 76, ...] is A002467. (End)
From Vaclav Kotesovec, Oct 13 2018: (Start)
Equals A185393 - 1.
Equals -LambertW(exp(1/(1 - exp(1))) / (1 - exp(1))).
Equals -1 - LambertW(-1, exp(1/(1 - exp(1))) / (1 - exp(1))). (End)
From Gleb Koloskov, Sep 03 2021: (Start)
Equals (coth(1/2)-1)/2 = (A307178-1)/2.
Equals 1/2 + 2*Integral_{x=0..oo} sin(x)/(exp(2*Pi*x)-1) dx.
Equals 1/2 + (1/Pi)*Integral_{x=0..1} sin(log(x)/(2*Pi))/(x-1) dx. (End)
Equals -lim_{n->oo} zeta(1-n, n)*n^(1 - n). - Vaclav Kotesovec and Peter Luschny, Nov 05 2021
Equals Integral_{x=0..1} floor(-log(x)) dx (see Redmond link). - Amiram Eldar, Oct 03 2023
Equals 1/2 + Sum_{k>=2} tanh(1/2^k)/2^k. - Antonio Graciá Llorente, Jan 21 2024

Entry revised by N. J. A. Sloane, Apr 07 2006

A356129 a(n) = Sum_{k=1..n} k * sigma_{n-1}(k).

Original entry on oeis.org

1, 7, 41, 395, 4503, 68969, 1205345, 24831145, 574932340, 14936279962, 427782949566, 13426887958078, 457622797727840, 16842616079514468, 665489067204502336, 28102162931539093732, 1262904299189373463930, 60182778247311758955112

Offset: 1

Seiichi Manyama, Jul 27 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..386

Cf. A356124, A356130, A356131.

a[n_] := Sum[k * DivisorSigma[n - 1, k], {k, 1, n}]; Array[a, 18] (* Amiram Eldar, Jul 28 2022 *)

PARI
```
a(n) = sum(k=1, n, k*sigma(k, n-1));
```

a(n) = sum(k=1, n, k^n*binomial(n\k+1, 2));

from math import isqrt
from sympy import bernoulli
def A356129(n): return ((s:=isqrt(n))*(s+1)*(bernoulli(n+1)-bernoulli(n+1,s+1))+sum(k**n*(n+1)*(q:=n//k)*(q+1)+(k*(bernoulli(n+1,q+1)-bernoulli(n+1))<<1) for k in range(1,s+1)))//(n+1)>>1 # Chai Wah Wu, Oct 24 2023

a(n) = Sum_{k=1..n} k^n * binomial(floor(n/k)+1,2).
a(n) = [x^n] (1/(1-x)) * Sum_{k>=1} k^n * x^k/(1 - x^k)^2.
a(n) ~ c * n^n, where c = 1/(1 - 1/exp(1)) = A185393. - Vaclav Kotesovec, Aug 07 2022

A349963 a(n) = Sum_{k=0..n} (2*k)^n.

Original entry on oeis.org

1, 2, 20, 288, 5664, 141600, 4298944, 153638912, 6319260672, 294044152320, 15272286131200, 875880428003328, 54976337351106560, 3748609104907476992, 275924407293425336320, 21806398621389422592000, 1841661678145084557099008, 165530736067119754944577536

Offset: 0

Seiichi Manyama, Dec 07 2021

nonn

Cf. A031971, A185393, A249459, A349970.

a[n_] := Sum[If[k == n == 0, 1, (2*k)^n], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Dec 07 2021 *)

PARI
```
a(n) = sum(k=0, n, (2*k)^n);
```

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (2*k*x)^k/(1-2*k*x)))

G.f.: Sum_{k>=0} (2*k * x)^k/(1 - 2*k * x).
a(n) = 2^n * A031971(n).
a(n) ~ c * 2^n * n^n, where c = 1/(1 - 1/exp(1)) = A185393. - Vaclav Kotesovec, Dec 07 2021

A356046 a(n) = Sum_{k=1..n} sigma_n(k) * floor(n/k).

Original entry on oeis.org

1, 7, 40, 393, 4498, 68898, 1205205, 24830617, 574911611, 14936215765, 427782762142, 13426870089265, 457622727372932, 16842615801316402, 665489035541044561, 28102162770144986248, 1262904298391426474369, 60182778141796948356895

Offset: 1

Seiichi Manyama, Jul 24 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..386

Cf. A321141, A356045.

PARI
```
a(n) = sum(k=1, n, sigma(k, n)*(n\k));
```

a(n) = sum(k=1, n, sumdiv(k, d, d^n*numdiv(k/d)));

a(n) = sum(k=1, n, sumdiv(k, d, sigma(d, n)));

a(n) = [x^n] (1/(1-x)) * Sum_{k>=1} sigma_n(k) * x^k/(1 - x^k).
a(n) = Sum_{k=1..n} Sum_{d|k} d^n * tau(k/d).
a(n) = Sum_{k=1..n} Sum_{d|k} sigma_n(d).
a(n) ~ c * n^n, where c = 1/(1 - 1/exp(1)) = A185393. - Vaclav Kotesovec, Aug 07 2022

A052809 A simple grammar: number of cycles of cycles.

Original entry on oeis.org

0, 0, 2, 6, 28, 175, 1368, 12838, 140656, 1762794, 24878320, 390495336, 6748280064, 127324033824, 2604355096224, 57404425654080, 1356401049662208, 34202807058719568, 916723959720053760

Offset: 0

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

Seiichi Manyama, Table of n, a(n) for n = 0..419
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 771

Cf. A003713, A185393.

spec := [S,{B=Cycle(C),C=Cycle(Z),S=Prod(B,Z)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

CoefficientList[Series[-Log[1+Log[1-x]]*x, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 01 2013 *)

a(n) = n*sum(k=1, n-1, (k-1)!*abs(stirling(n-1, k, 1))); \\ Seiichi Manyama, May 20 2022

E.g.f.: log(-1/(-1+log(-1/(-1+x))))*x.
E.g.f.: -log(1+log(1-x))*x. - Vaclav Kotesovec, Oct 01 2013
a(n) ~ (n-1)! * (exp(1)/(exp(1)-1))^(n-1). - Vaclav Kotesovec, Oct 01 2013
a(n) = n * Sum_{k=1..n-1} (k-1)! * |Stirling1(n-1,k)| = n * A003713(n-1). - Seiichi Manyama, May 20 2022

A254445 Decimal expansion of Sum_{n>=0} 1/cosh(n).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Jan 30 2015

			2.071121329967823169571572992683526260893794658306930755815650661114451...

Cf. A185393, A254446, A335415.

evalf(sum(1/cosh(n), n = 0..infinity), 120);

RealDigits[N[Re[Sum[1/Cosh[n],{n,0,Infinity}]],110],10,105][[1]]

suminf(n=0, 1/cosh(n)) \\ Michel Marcus, Jan 30 2015

A254446 Decimal expansion of Sum_{n>=1} 1/sinh(n).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Jan 30 2015

			1.284423027303676524572857579841338858444029376554223611185650493880957...

Cf. A185393, A240964, A254445, A335414.

evalf(sum(1/sinh(n), n = 1..infinity), 120);

RealDigits[N[Re[Sum[1/Sinh[n],{n,1,Infinity}]],110],10,105][[1]]

suminf(n=1, 1/sinh(n)) \\ Michel Marcus, Jan 30 2015

cp's OEIS Frontend

A063048 Numbers n such that the Reverse and Add! trajectory of n (presumably) does not reach a palindrome and does not join the trajectory of any term m < n.

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A068996 Decimal expansion of 1 - 1/e.

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A086787 a(n) = Sum_{i=1..n} ( Sum_{j=1..n} i^j ).

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A073333 Decimal expansion of 1/(e - 1) = Sum_{k >= 1} exp(-k).

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A356129 a(n) = Sum_{k=1..n} k * sigma_{n-1}(k).

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A349963 a(n) = Sum_{k=0..n} (2*k)^n.

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