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).

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

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

T. D. Noe, Table of n, a(n) for n = 0..201

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, 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.

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

Adjacent sequences: A113871 A113872 A113873 * A113875 A113876 A113877

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