A003417 -id:A003417 - cp's OEIS frontend

A001113 Decimal expansion of e.

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

Offset: 1

N. J. A. Sloane

e is sometimes called Euler's number or Napier's constant.
Also, decimal expansion of sinh(1)+cosh(1). - Mohammad K. Azarian, Aug 15 2006
If m and n are any integers with n > 1, then |e - m/n| > 1/(S(n)+1)!, where S(n) = A002034(n) is the smallest number such that n divides S(n)!. - Jonathan Sondow, Sep 04 2006
Limit_{n->infinity} A000166(n)*e - A000142(n) = 0. - Seiichi Kirikami, Oct 12 2011
Euler's constant (also known as Euler-Mascheroni constant) is gamma = 0.57721... and Euler's number is e = 2.71828... . - Mohammad K. Azarian, Dec 29 2011
One of the many continued fraction expressions for e is 2+2/(2+3/(3+4/(4+5/(5+6/(6+ ... from Ramanujan (1887-1920). - Robert G. Wilson v, Jul 16 2012
e maximizes the value of x^(c/x) for any real positive constant c, and minimizes for it for a negative constant, on the range x > 0. This explains why elements of A000792 are composed primarily of factors of 3, and where needed, some factors of 2. These are the two primes closest to e. - Richard R. Forberg, Oct 19 2014
There are two real solutions x to c^x = x^c when c, x > 0 and c != e, one of which is x = c, and only one real solution when c = e, where the solution is x = e. - Richard R. Forberg, Oct 22 2014
This is the expected value of the number of real numbers that are independently and uniformly chosen at random from the interval (0, 1) until their sum exceeds 1 (Bush, 1961). - Amiram Eldar, Jul 21 2020

			2.71828182845904523536028747135266249775724709369995957496696762772407663...

Florian Cajori, A History of Mathematical Notations, Dover edition (2012), par. 400.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 24, 250-256.
Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.3.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.4 Irrational Numbers, p. 85.
E. Maor, e: The Story of a Number, Princeton Univ. Press, 1994.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 52.
G. W. Reitwiesner, An ENIAC determination of pi and e to more than 2000 decimal places. Math. Tables and Other Aids to Computation 4, (1950). 11-15.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapters 1 and 2, equations 1:7:4, 2:5:4 at pages 13, 20.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 46.

N. J. A. Sloane, Table of 50000 digits of e labeled from 1 to 50000 [based on the ICON Project link below]
Mohammad K. Azarian, An Expansion of e, Problem # B-765, Fibonacci Quarterly, Vol. 32, No. 2, May 1994, p. 181. Solution appeared in Vol. 33, No. 4, Aug. 1995, p. 377.
Mohammad K. Azarian, Euler's Number Via Difference Equations, International Journal of Contemporary Mathematical Sciences, Vol. 7, 2012, No. 22, pp. 1095 - 1102.
L. E. Bush, The William Lowell Putnam Mathematical Competition, The American Mathematical Monthly, Vol. 68, No. 1 (1961), pp. 18-33, problem 3.
Ed Copeland and Brady Haran, A proof that e is irrational, Numberphile video (2021).
Dave's Math Tables, e
X. Gourdon, Plouffe's Inverter, e to 1.250 billion digits
X. Gourdon and P. Sebah, The constant e and its computation
Brady Haran and James Grime, Incredible Formula, Numberphile YouTube video, 2016.
ICON Project, e to 50000 places
Roger Mansuy, Un intrigant poème... mathématique, Images des Mathématiques, CNRS, 2023. In French.
R. Nemiroff and J. Bonnell, The first 5 million digits of the number e
Remco Niemeijer, Digits Of E, programmingpraxis
J. J. O'Connor & E. F. Robertson, The number e
Michael Penn, e is irrational, YouTube video, 2020.
Simon Plouffe, A million digits
G. W. Reitwiesner, An ENIAC determination of pi and e to more than 2000 decimal places, Pi, A Source book, pp 277-281, 2000.
E. Sandifer, How Euler Did It, Who proved e is irrational?, MAA Online (2006)
D. Shanks and J. W. Wrench, Jr., Calculation of e to 100,000 decimals, Math. Comp., 23 (1969), 679-680.
Jean-Louis Sigrist, Le premier million de décimales de e.
Vladimir Ivanovich Smirnov, A course of higher mathematics, vol. 1 , Pergamon Press, 1964, p. 339.
Jonathan Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly, 113 (2006), 637-641 (article) and 114 (2007), 659 (addendum).
Jonathan Sondow and K. Schalm, Which partial sums of the Taylor series for e are convergents to e? (and a link to the primes 2, 5, 13, 37, 463), II, Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010.
G. Villemin's Almanach of Numbers, Constant "e"
Eric Weisstein's World of Mathematics, e
Eric Weisstein's World of Mathematics, e Digits
Eric Weisstein's World of Mathematics, Factorial Sums
Eric Weisstein's World of Mathematics, Uniform Sum Distribution
Eric Weisstein's World of Mathematics, e Approximations
Wikipedia, E (mathematical constant)
Index entries for "core" sequences
Index entries for transcendental numbers

Cf. A002034, A003417 (continued fraction), A073229, A122214, A122215, A122216, A122217, A122416, A122417.
Expansion of e in base b: A004593 (b=2), A004594 (b=3), A004595 (b=4), A004596 (b=5), A004597 (b=6), A004598 (b=7), A004599 (b=8), A004600 (b=9), this sequence (b=10), A170873 (b=16). - Jason Kimberley, Dec 05 2012
Powers e^k: A092578 (k = -7), A092577 (k = -6), A092560 (k = -5), A092553 - A092555 (k = -2 to -4), A068985 (k = -1), A072334 (k = 2), A091933 (k = 3), A092426 (k = 4), A092511 - A092513 (k = 5 to 7).

-- See Niemeijer link.
a001113 n = a001113_list !! (n-1)
a001113_list = eStream (1, 0, 1)
   [(n, a * d, d) | (n, d, a) <- map (\k -> (1, k, 1)) [1..]] where
   eStream z xs'@(x:xs)
     | lb /= approx z 2 = eStream (mult z x) xs
     | otherwise = lb : eStream (mult (10, -10 * lb, 1) z) xs'
     where lb = approx z 1
           approx (a, b, c) n = div (a * n + b) c
           mult (a, b, c) (d, e, f) = (a * d, a * e + b * f, c * f)
-- Reinhard Zumkeller, Jun 12 2013

Digits := 200: it := evalf((exp(1))/10, 200): for i from 1 to 200 do printf(`%d,`,floor(10*it)): it := 10*it-floor(10*it): od: # James Sellers, Feb 13 2001

RealDigits[E, 10, 120][[1]] (* Harvey P. Dale, Nov 14 2011 *)

e = Sum_{k >= 0} 1/k! = lim_{x -> 0} (1+x)^(1/x).
e is the unique positive root of the equation Integral_{u = 1..x} du/u = 1.
exp(1) = ((16/31)*(1 + Sum_{n>=1} ((1/2)^n*((1/2)*n^3 + (1/2)*n + 1)/n!)))^2. Robert Israel confirmed that the above formula is correct, saying: "In fact, Sum_{n=0..oo} n^j*t^n/n! = P_j(t)*exp(t) where P_0(t) = 1 and for j >= 1, P_j(t) = t (P_(j-1)'(t) + P_(j-1)(t)). Your sum is 1/2*P_3(1/2) + 1/2*P_1(1/2) + P_0(1/2)." - Alexander R. Povolotsky, Jan 04 2009
exp(1) = (1 + Sum_{n>=1} ((1+n+n^3)/n!))/7. - Alexander R. Povolotsky, Sep 14 2011
e = 1 + (2 + (3 + (4 + ...)/4)/3)/2 = 2 + (1 + (1 + (1 + ...)/4)/3)/2. - Rok Cestnik, Jan 19 2017
From Peter Bala, Nov 13 2019: (Start)
The series representation e = Sum_{k >= 0} 1/k! is the case n = 0 of the more general result e = n!*Sum_{k >= 0} 1/(k!*R(n,k)*R(n,k+1)), n = 0,2,3,4,..., where R(n,x) is the n-th row polynomial of A269953.
e = 2 + Sum_{n >= 0} (-1)^n*(n+2)!/(d(n+2)*d(n+3)), where d(n) = A000166(n).
e = Sum_{n >= 0} (x^2 + (n+2)*x + n)/(n!(n + x)*(n + 1 + x)), provided x is not zero or a negative integer. (End)
Equals lim_{n -> oo} (2*3*5*...*prime(n))^(1/prime(n)). - Peter Luschny, May 21 2020
e = 3 - Sum_{n >= 0} 1/((n+1)^2*(n+2)^2*n!). - Peter Bala, Jan 13 2022
e = lim_{n->oo} prime(n)*(1 - 1/n)^prime(n). - Thomas Ordowski, Jan 31 2023
e = 1+(1/1)*(1+(1/2)*(1+(1/3)*(1+(1/4)*(1+(1/5)*(1+(1/6)*(...)))))), equivalent to the first formula. - David Ulgenes, Dec 01 2023
From Michal Paulovic, Dec 12 2023: (Start)
Equals lim_{n->oo} (1 + 1/n)^n.
Equals x^(x^(x^...)) (infinite power tower) where x = e^(1/e) = A073229. (End)
Equals Product_{k>=1} (1 + 1/k) * (1 - 1/(k + 1)^2)^k. - Antonio Graciá Llorente, May 14 2024
Equals lim_{n->oo} Product_{k=1..n} (n^2 + k)/(n^2 - k) (see Finch). - Stefano Spezia, Oct 19 2024
e ~ (1 + 9^((-4)^(7*6)))^(3^(2^85)), correct to more than 18*10^24 digits (Richard Sabey, 2004); see Haran and Grime link. - Paolo Xausa, Dec 21 2024.

A007677 Denominators of convergents to e.

Original entry on oeis.org

1, 1, 3, 4, 7, 32, 39, 71, 465, 536, 1001, 8544, 9545, 18089, 190435, 208524, 398959, 4996032, 5394991, 10391023, 150869313, 161260336, 312129649, 5155334720, 5467464369, 10622799089, 196677847971, 207300647060, 403978495031, 8286870547680, 8690849042711

Offset: 0

N. J. A. Sloane, Robert G. Wilson v

Same as A113874 without its first two terms. - Jonathan Sondow, Aug 16 2006

			2, 3, 8/3, 11/4, 19/7, 87/32, 106/39, 193/71, 1264/465, 1457/536, 2721/1001, 23225/8544, 25946/9545, 49171/18089, 517656/190435, 566827/208524, 1084483/398959, 13580623/4996032, 14665106/5394991, 28245729/10391023, 410105312/150869313, 438351041/161260336, 848456353/312129649, ...

CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 88.
W. J. LeVeque, Fundamentals of Number Theory. Addison-Wesley, Reading, MA, 1977, p. 240.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Eric W. Weisstein, Table of n, a(n) for n = 0..1000 (first 200 terms from T. D. Noe)
L. Bayon, P. Fortuny, J. M. Grau, M. M. Ruiz, M. A. Oller-Marcen, The Best-or-Worst and the Postdoc problems with random number of candidates, arXiv:1809.06390 [math.PR], 2018.
E. B. Burger, Diophantine Olympics and World Champions: Polynomials and Primes Down Under, Amer. Math. Monthly, 107 (Nov. 2000), 822-829.
C. Elsner, Series of Error Terms for Rational Approximations of Irrational Numbers, J. Int. Seq. 14 (2011) # 11.1.4.
C. Elsner, M. Stein, On Error Sum Functions Formed by Convergents of Real Numbers, J. Int. Seq. 14 (2011) # 11.8.6
J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly 113 (2006) 637-641 (article), 114 (2007) 659 (addendum).
J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, arXiv:0704.1282 [math.HO], 2007-2010.
J. Sondow and K. Schalm, Which partial sums of the Taylor series for e are convergents to e? (and a link to the primes 2, 5, 13, 37, 463), II, Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010; arXiv:0709.0671 [math.NT], 2007-2009.
Eric Weisstein's World of Mathematics, e Continued Fraction
Eric Weisstein's World of Mathematics, Sultan's Dowry Problem

Cf. A007676 (numerators of convergents to e).

Cf. A003417 (continued fraction of e).

Digits := 60: E := exp(1); convert(evalf(E),confrac,50,'cvgts'): cvgts;

Denominator[Convergents[E, 40]] (* T. D. Noe, Oct 12 2011 *)
Denominator[Table[Piecewise[{
   {Hypergeometric1F1[-1 - n/3, -1 - (2 n)/3, 1]/Hypergeometric1F1[-(n/3), -1 - (2 n)/3, -1], Mod[n, 3] == 0},
   {Hypergeometric1F1[1/3 (-2 - n), -(2/3) (2 + n), 1]/Hypergeometric1F1[1/3 (-2 - n), -(2/3) (2 + n), -1], Mod[n, 3] == 1},
   {Hypergeometric1F1[1/3 (-1 - n), 1 - (2 (4 + n))/3, 1]/Hypergeometric1F1[1/3 (-4 - n), 1 - (2 (4 + n))/3, -1], Mod[n, 3] == 2}
}], {n, 0, 30}]] (* Eric W. Weisstein, Sep 09 2013 *)
Table[Piecewise[{
    {(1 + (2 n)/3)!/(n/3)! Hypergeometric1F1[-(n/3), -1 - (2 n)/3, -1], Mod[n, 3] == 0},
    {((2 (2 + n))/3)!/((2 + n)/3)! Hypergeometric1F1[1/3 (-2 - n), -(2/3) (2 + n), -1], Mod[n, 3] == 1},
    {((4 + n) (5/3 + (2 n)/3)! )/(3 ((4 + n)/3)!) Hypergeometric1F1[1/3 (-4 - n), 1 - (2 (4 + n))/3, -1], Mod[n, 3] == 2}
}], {n, 0, 30}]  (* Eric W. Weisstein, Sep 10 2013 *)

A007676 Numerators of convergents to e.

Original entry on oeis.org

2, 3, 8, 11, 19, 87, 106, 193, 1264, 1457, 2721, 23225, 25946, 49171, 517656, 566827, 1084483, 13580623, 14665106, 28245729, 410105312, 438351041, 848456353, 14013652689, 14862109042, 28875761731, 534625820200, 563501581931, 1098127402131, 22526049624551

Offset: 0

N. J. A. Sloane, Robert G. Wilson v

Same as A113873 without its first two terms. - Jonathan Sondow, Aug 16 2006

			2, 3, 8/3, 11/4, 19/7, 87/32, 106/39, 193/71, 1264/465, 1457/536, 2721/1001, 23225/8544, 25946/9545, 49171/18089, 517656/190435, 566827/208524, 1084483/398959, 13580623/4996032, 14665106/5394991, 28245729/10391023, 410105312/150869313, 438351041/161260336, 848456353/312129649, ...

CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 88.
W. J. LeVeque, Fundamentals of Number Theory. Addison-Wesley, Reading, MA, 1977, p. 240.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Eric W. Weisstein, Table of n, a(n) for n = 0..1000 (first 200 terms from T. D. Noe)
L. Bayon, P. Fortuny, J. M. Grau, M. M. Ruiz, M. A. Oller-Marcen, The Best-or-Worst and the Postdoc problems with random number of candidates, arXiv:1809.06390 [math.PR], 2018.
C. Elsner, Series of Error Terms for Rational Approximations of Irrational Numbers, J. Int. Seq. 14 (2011) # 11.1.4.
C. Elsner, M. Stein, On Error Sum Functions Formed by Convergents of Real Numbers, J. Int. Seq. 14 (2011) # 11.8.6.
J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly 113 (2006) 637-641.
J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, arXiv:0704.1282 [math.HO], 2007-2010.
J. Sondow and K. Schalm, Which partial sums of the Taylor series for e are convergents to e? (and a link to the primes 2, 5, 13, 37, 463), II, Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010.
Eric Weisstein's World of Mathematics, e Continued Fraction
Eric Weisstein's World of Mathematics, Sultan's Dowry Problem

Cf. A007677 (denominators of convergents to e).

Cf. A003417 (continued fraction of e).

Digits := 60: convert(evalf(E),confrac,50,'cvgts'): cvgts;

Numerator[Convergents[E, 30]] (* T. D. Noe, Oct 12 2011 *)
Numerator[Table[Piecewise[{
   {Hypergeometric1F1[-1 - n/3, -1 - (2 n)/3, 1]/Hypergeometric1F1[-(n/3), -1 - (2 n)/3, -1], Mod[n, 3] == 0},
   {Hypergeometric1F1[1/3 (-2 - n), -(2/3) (2 + n), 1]/Hypergeometric1F1[1/3 (-2 - n), -(2/3) (2 + n), -1], Mod[n, 3] == 1},
   {Hypergeometric1F1[1/3 (-1 - n), 1 - (2 (4 + n))/3, 1]/Hypergeometric1F1[1/3 (-4 - n), 1 - (2 (4 + n))/3, -1], Mod[n, 3] == 2}
}], {n, 0, 30}]] (* Eric W. Weisstein, Sep 09 2013 *)
Table[Piecewise[{
   {(-1 + (2 (3 + n))/3)!/(-1 + (3 + n)/3)! Hypergeometric1F1[1/3 (-3 - n), 1 - (2 (3 + n))/3, 1], Mod[n, 3] == 0},
   {((2 (2 + n))/3)!/((2 + n)/3)! Hypergeometric1F1[1/3 (-2 - n), -(2/3) (2 + n), 1], Mod[n, 3] == 1},
   {(5/3 + (2 n)/3)!/((1 + n)/3)! Hypergeometric1F1[1/3 (-1 - n), 1 - (2 (4 + n))/3, 1], Mod[n, 3] == 2}
}], {n, 0, 30}] (* Eric W. Weisstein, Sep 10 2013 *)

A005131 A generalized continued fraction for Euler's number e.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, 1, 1, 14, 1, 1, 16, 1, 1, 18, 1, 1, 20, 1, 1, 22, 1, 1, 24, 1, 1, 26, 1, 1, 28, 1, 1, 30, 1, 1, 32, 1, 1, 34, 1, 1, 36, 1, 1, 38, 1, 1, 40, 1, 1, 42

Offset: 0

Russ Cox

Only a(1) = 0 prevents this from being a simple continued fraction. The motivation for this alternate representation is that the simple pattern {1, 2*n, 1} (from n=0) may be more mathematically appealing than the pattern in the corresponding simple continued fraction (at A003417). - Joseph Biberstine (jrbibers(AT)indiana.edu), Aug 14 2006

If we consider a(n) = A005131(n+1), n >= 0, then we get the simple continued fraction for 1/(e-1). - Daniel Forgues, Apr 19 2011

Douglas Hofstadter, "Fluid Concepts and Creative Analogies: Computer Models of the Fundamental Mechanisms of Thought".

N. J. A. Sloane, Table of n, a(n) for n = 0..500
H. Cohn, A short proof of the simple continued fraction expansion of e, Amer. Math. Monthly, 113 (No. 1, 2006), 57-62. [JSTOR] and arXiv:math/0601660.
D. H. Lehmer, Continued fractions containing arithmetic progressions, Scripta Mathematica, 29 (1973): 17-24. [Annotated copy of offprint]
T. J. Osler, A proof of the continued fraction expansion of e^(1/M), Amer. Math. Monthly, 113 (No. 1, 2006), 62-66.
A. J. van der Poorten, Continued fraction expansions of values of the exponential function...
A. J. van der Poorten, Number theory and Kustaa Inkeri

Cf. A003417, A100261.

Table[If[Mod[k, 3] == 1, 2/3*(k - 1), 1], {k, 0, 80}] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Aug 14 2006 *)

a(n)=if(n>=0,[1, 2*(n\3), 1][n%3+1]) \\ Jaume Oliver Lafont, Nov 14 2009

If n==1 (mod 3), then a(n) = 2*(n-1)/3, otherwise a(n) = 1. - Joseph Biberstine (jrbibers(AT)indiana.edu), Aug 14 2006
G.f. = (-x^5 + 2*x^4 - x^3 + x^2 + 1)/(x^6 - 2*x^3 + 1). - Alexander R. Povolotsky, Apr 26 2008
{-a(n)-2*a(n+1)-3*a(n+2)-2*a(n+3)-a(n+4)+2*n+8, a(0) = 1, a(1) = 0, a(2) = 1, a(3) = 1, a(4) = 2, a(5) = 1}. - Robert Israel, May 14 2008
a(n) = 1 + 2*(2*n-5) * (cos(2*Pi*(n-1)/3)+1/2)/9. - David Spitzer, Jan 09 2017

Edited by M. F. Hasler, Jan 26 2014

A056072 a(n) = floor(e^e^ ... ^e), with n e's.

Original entry on oeis.org

1, 2, 15, 3814279

Offset: 0

Robert G. Wilson v, Jul 26 2000

nonn

The next term is too large to include.
From Vladimir Reshetnikov, Apr 27 2013: (Start)
a(4) = 2331504399007195462289689911...2579139884667434294745087021 (1656521 decimal digits in total), given by initial segment of A085667.
a(5) has more than 10^10^6 decimal digits.
a(6) has more than 10^10^10^6 decimal digits. (End)

Cf. A001113, A004002, A003417, A064107, A159825, A073226, A073227, A085667, A096232, A225064, A225053.

p:= n-> `if`(n=0, 1, exp(1)^p(n-1)):
a:= n-> floor(p(n)):
seq(a(n), n=0..3);  # Alois P. Heinz, Jul 20 2024

Floor[NestList[Exp, 1, 3]] (* Vladimir Reshetnikov, Apr 29 2013 *)

A056072(n,f=floor)=f(exp(if(n>0,A056072(n-1,x->x))))  \\ M. F. Hasler, May 01 2013

A001204 Continued fraction for e^2.

Original entry on oeis.org

7, 2, 1, 1, 3, 18, 5, 1, 1, 6, 30, 8, 1, 1, 9, 42, 11, 1, 1, 12, 54, 14, 1, 1, 15, 66, 17, 1, 1, 18, 78, 20, 1, 1, 21, 90, 23, 1, 1, 24, 102, 26, 1, 1, 27, 114, 29, 1, 1, 30, 126, 32, 1, 1, 33, 138, 35, 1, 1, 36, 150, 38, 1, 1, 39, 162, 41, 1, 1, 42, 174, 44, 1, 1, 45, 186, 47, 1, 1

Offset: 0

N. J. A. Sloane

Note that e^2 = 7 + 2/(5 + 1/(7 + 1/(9 + 1/(11 + ...)))) (follows from the fact that A004273 is the continued fraction expansion of tanh(1) = (e^2 - 1)/(e^2 + 1)). - Peter Bala, Jan 15 2022

			7.389056098930650227230427460... = 7 + 1/(2 + 1/(1 + 1/(1 + 1/(3 + ...)))).

Oskar Perron, Die Lehre von den Kettenbrüchen, 2nd ed., Teubner, Leipzig, 1929, p. 138.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Harry J. Smith, Table of n, a(n) for n = 0..20000
Keith Matthews, Finding the continued fraction of e^(l/m).
Gang Xiao, Contfrac.
Index entries for continued fractions for constants.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,2,0,0,0,0,-1).

Cf. A003417, A004273, A005131, A058282, A072334.

ContinuedFraction[ E^2, 100]
LinearRecurrence[{0,0,0,0,2,0,0,0,0,-1},{7,2,1,1,3,18,5,1,1,6,30},80] (* Harvey P. Dale, Dec 30 2023 *)

PARI
```
contfrac(exp(2))
```

allocatemem(932245000); default(realprecision, 95000); x=contfrac(exp(2)); for (n=1, 20001, write("b001204.txt", n-1, " ", x[n])); \\ Harry J. Smith, Apr 30 2009

G.f.: (x^10 - x^8 - x^7 + x^6 + 4x^5 + 3x^4 + x^3 + x^2 + 2x + 7)/(x^5 - 1)^2. - Ralf Stephan, Mar 23 2003
For n > 0, a(5n) = 12n + 6, a(5n+1) = 3n + 2, a(5n+2) = a(5n+3) = 1 and a(5n+4) = 3n + 3. - Dean Hickerson, Mar 25 2003
Sum_{n>=5} (-1)^(n+1)/a(n) = (8*sqrt(3)-3)*Pi/72 - 2*log(2)/3. - Amiram Eldar, May 04 2025

More terms from Robert G. Wilson v, Dec 07 2000

cp's OEIS Frontend

A080408 Boustrophedon transform of the continued fraction of e (A003417).

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A080409 Decimal expansion of the number which results when the Boustrophedon transform of the continued fraction of e (A080408, A003417) is interpreted as a continued fraction.

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A001113 Decimal expansion of e.

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A007677 Denominators of convergents to e.

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A007676 Numerators of convergents to e.

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A005131 A generalized continued fraction for Euler's number e.

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A056072 a(n) = floor(e^e^ ... ^e), with n e's.

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