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A007676 Numerators of convergents to e.
(Formerly M0869)
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
Same as A113873 without its first two terms. - Jonathan Sondow, Aug 16 2006
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
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, 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 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
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}
}], {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 *)
CROSSREFS
Cf. A007677 (denominators of convergents to e).
Cf. A003417 (continued fraction of e).
Sequence in context: A206241 A295333 A113873 * A042443 A042263 A280323
KEYWORD
nonn,easy,nice,frac
AUTHOR
STATUS
approved

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Last modified March 19 03:33 EDT 2024. Contains 370952 sequences. (Running on oeis4.)