Displaying 1-10 of 36 results found.
Boustrophedon transform of the continued fraction of e ( A003417).
+20
1
2, 3, 6, 14, 35, 116, 448, 1980, 10098, 57840, 368201, 2578384, 19697486, 163017000, 1452918806, 13874348700, 141322966623, 1529472867448, 17526468199148, 211996227034964, 2699219798770446, 36085910558435148, 505406091697374877
LINKS
J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J.Combin. Theory, 17A (1996) 44-54 ( Abstract, pdf, ps).
FORMULA
a(n) appears to be asymptotic to C*n!*(2/Pi)^n where C = 9.27921365277635263761227970562207183019110298580498662908878310... - Benoit Cloitre and Mark Hudson (mrmarkhudson(AT)hotmail.com)
EXAMPLE
We simply apply the Boustrophedon transform to [2,1,2,1,1,4,1,1,6,1,1,8,1,1,...]
PROG
(Python)
from itertools import count, islice, accumulate
def A080408_gen(): # generator of terms blist = tuple()
for n in count(1):
yield (blist := tuple(accumulate(reversed(blist), initial=2 if n == 1 else 1 if n % 3 else n//3<<1)))[-1]
AUTHOR
Mark Hudson (mrmarkhudson(AT)hotmail.com), Feb 17 2003
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.
+20
1
2, 3, 1, 5, 9, 8, 4, 7, 3, 6, 1, 5, 7, 8, 9, 1, 3, 8, 3, 3, 7, 8, 5, 9, 5, 3, 5, 0, 9, 2, 0, 4, 0, 6, 8, 1, 6, 1, 7, 4, 4, 9, 6, 8, 5, 7, 3, 8, 1, 3, 5, 7, 7, 6, 4, 3, 4, 2, 2, 0, 8, 1, 7, 7, 1, 2, 1, 0, 1, 9, 5, 9, 8, 7, 2, 8, 7, 6, 9, 0, 1, 2, 7, 4, 5, 7, 1, 9, 8, 7, 0, 1, 3, 2, 2, 3, 8, 5, 3, 5
LINKS
J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A (1996) 44-54 ( Abstract, pdf, ps).
EXAMPLE
2.315984736157891383378595350920406816174496857381357764342208...
AUTHOR
Mark Hudson (mrmarkhudson(AT)hotmail.com), Feb 17 2003
Decimal expansion of e.
(Formerly M1727 N0684)
+10
661
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
COMMENTS
e is sometimes called Euler's number or Napier's constant.
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 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
REFERENCES
S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.3.
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).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 46.
LINKS
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. 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, eEric Weisstein's World of Mathematics, e Digits
FORMULA
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 e = 1 + (2 + (3 + (4 + ...)/4)/3)/2 = 2 + (1 + (1 + (1 + ...)/4)/3)/2. - Rok Cestnik, Jan 19 2017 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 Equals lim_{n->oo} (1 + 1/n)^n.
Equals x^(x^(x^...)) (infinite power tower) where x = e^(1/e) = A073229. (End) Equals lim_{n->oo} Product_{k=1..n} (n^2 + k)/(n^2 - k) (see Finch). - Stefano Spezia, Oct 19 2024
EXAMPLE
2.71828182845904523536028747135266249775724709369995957496696762772407663...
MAPLE
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 A. Sellers, Feb 13 2001
PROG
(PARI) default(realprecision, 50080); x=exp(1); for (n=1, 50000, d=floor(x); x=(x-d)*10; write("b001113.txt", n, " ", d)); \\ Harry J. Smith, Apr 15 2009 (Haskell) -- 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)
CROSSREFS
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
Denominators of convergents to e.
(Formerly M2343)
+10
34
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
REFERENCES
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).
LINKS
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.
EXAMPLE
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, ...
MAPLE
Digits := 60: E := exp(1); convert(evalf(E), confrac, 50, 'cvgts'): cvgts;
MATHEMATICA
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}
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}
CROSSREFS
Cf. A007676 (numerators of convergents to e). Cf. A003417 (continued fraction of e).
Numerators of convergents to e.
(Formerly M0869)
+10
28
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
REFERENCES
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).
LINKS
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.
EXAMPLE
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, ...
MAPLE
Digits := 60: convert(evalf(E), confrac, 50, 'cvgts'): cvgts;
MATHEMATICA
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}
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}
CROSSREFS
Cf. A007677 (denominators of convergents to e). Cf. A003417 (continued fraction of e).
A generalized continued fraction for Euler's number e.
+10
7
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
COMMENTS
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
REFERENCES
Douglas Hofstadter, "Fluid Concepts and Creative Analogies: Computer Models of the Fundamental Mechanisms of Thought".
LINKS
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.
FORMULA
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
{-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
MATHEMATICA
Table[If[Mod[k, 3] == 1, 2/3*(k - 1), 1], {k, 0, 80}] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Aug 14 2006 *)
Continued fraction for e^2.
(Formerly M4322 N1811)
+10
6
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
COMMENTS
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
REFERENCES
O. 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).
LINKS
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 2, 0, 0, 0, 0, -1).
FORMULA
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
EXAMPLE
7.389056098930650227230427460... = 7 + 1/(2 + 1/(1 + 1/(1 + 1/(3 + ...)))).
MATHEMATICA
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 *)
PROG
(PARI) contfrac(exp(2))
(PARI) 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
a(n) = floor(e^e^ ... ^e), with n e's.
+10
6
COMMENTS
The next term is too large to include.
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)
MAPLE
p:= n-> `if`(n=0, 1, exp(1)^p(n-1)):
a:= n-> floor(p(n)):
"Exact" continued fraction of e.
+10
6
3, -4, 2, 5, -2, -7, 2, 9, -2, -11, 2, 13, -2, -15, 2, 17, -2, -19, 2, 21, -2, -23, 2, 25, -2, -27, 2, 29, -2, -31, 2, 33, -2, -35, 2, 37, -2, -39, 2, 41, -2, -43, 2, 45, -2, -47, 2, 49, -2, -51, 2, 53, -2, -55, 2, 57, -2, -59, 2, 61, -2, -63, 2, 65, -2, -67, 2, 69, -2, -71, 2, 73, -2, -75, 2, 77, -2, -79, 2, 81, -2, -83, 2, 85, -2, -87, 2
COMMENTS
See comments in A133593. Just as for the usual continued fraction for e, the exact continued fraction also has a simple pattern.
FORMULA
x(0) = e, a(n) = floor( |x(n)| + 0.5 ) * Sign(x(n)), x(n+1) = 1 / (x(n)-a(n)).
a(n) = 1/2*((2-i*2)*((-i)^n-i*i^n)+((-i)^n-i^n)*n)*(-1)*i for n>1.
a(n) = -2*a(n-2)-a(n-4) for n>5.
G.f.: -(x^5-5*x^4+3*x^3-8*x^2+4*x-3)/(x^2+1)^2.
(End)
MATHEMATICA
$MaxExtraPrecision = Infinity; x[0] = E; a[n_] := a[n] = Round[Abs[x[n]]]*Sign[x[n]]; x[n_] := 1/(x[n - 1] - a[n - 1]); Table[a[n], {n, 0, 120}] (* Clark Kimberling, Sep 04 2013 *) Join[{3, -4}, LinearRecurrence[{0, -2, 0, -1}, {2, 5, -2, -7}, 100]] (* Vincenzo Librandi, Jan 09 2016 *)
PROG
(PARI) Vec(-(x^5-5*x^4+3*x^3-8*x^2+4*x-3)/(x^2+1)^2 + O(x^100)) \\ Colin Barker, Sep 13 2013
AUTHOR
Serhat Sevki Dincer (jfcgauss(AT)gmail.com), Dec 30 2007
Continued fraction for e^e^e A073227.
+10
5
3814279, 9, 1, 1, 4, 1, 53, 26, 1, 13, 3, 1, 1, 22, 1, 226, 1, 5, 2, 1, 6, 2, 3, 1, 4, 1, 6, 39, 2, 1, 3, 1, 5, 1, 4, 1, 3, 1, 4, 1, 1, 19, 1, 2, 8899, 5, 2, 2, 1, 3, 3, 2, 2, 2, 1, 1, 3, 5, 1, 6, 10, 2, 1, 2, 1, 1, 1, 2, 2, 4, 1, 10, 2, 6, 1, 5, 6, 2, 4, 2, 1, 2, 1, 1, 1, 3, 2, 2, 1, 1, 11, 7, 3, 1, 4, 4
COMMENTS
It was conjectured (but remains unproved) that this sequence is infinite and aperiodic, but it is difficult to determine who first posed this problem. - Vladimir Reshetnikov, Apr 27 2013
EXAMPLE
3814279.104760220592209... = 3814279 + 1/(9 + 1/(1 + 1/(1 + 1/(4 + ...)))).
PROG
(PARI) { allocatemem(932245000); default(realprecision, 21000); x=contfrac(exp(exp(exp(1)))); for (n=1, 20001, write("b159825.txt", n-1, " ", x[n])); }
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