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A113874 a(3n) = a(3n-1) + a(3n-2), a(3n+1) = 2n*a(3n) + a(3n-1), a(3n+2) = a(3n+1) + a(3n). 3
1, 0, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
A113873(n)/a(n) -> e.
Without the first two terms, same as A007677 (denominators of convergents to e). - Jonathan Sondow, Aug 16 2006
LINKS
H. Cohn, A short proof of the simple continued fraction expansion of e, Amer. Math. Monthly, 113 (No. 1, 2006), 57-62.
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.
MAPLE
a[0]:=1: a[1]:=0: a[2]:=1: for n from 3 to 34 do if n mod 3 = 0 then a[n]:=a[n-1]+a[n-2] elif n mod 3 = 1 then a[n]:=2*(n-1)*a[n-1]/3+a[n-2] else a[n]:=a[n-1]+a[n-2] fi: od: seq(a[n], n=0..34); # Emeric Deutsch, Jan 28 2006
MATHEMATICA
a[0] = 1; a[1] = 0; a[n_] := a[n] = Switch[ Mod[n, 3], 0, a[n - 1] + a[n - 2], 1, 2(n - 1)/3*a[n - 1] + a[n - 2], 2, a[n - 1] + a[n - 2]]; a /@ Range[0, 30]
Join[{1, 0}, Denominator[Convergents[E, 30]]] (* Harvey P. Dale, Aug 09 2014 *)
CROSSREFS
Cf. A113873.
Sequence in context: A041091 A270373 A117764 * A007677 A042773 A042173
KEYWORD
easy,nonn
AUTHOR
N. J. A. Sloane, Jan 27 2006
EXTENSIONS
More terms from Robert G. Wilson v and Emeric Deutsch, Jan 28 2006
STATUS
approved

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Last modified April 15 20:47 EDT 2024. Contains 371696 sequences. (Running on oeis4.)