A113873 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, 1, 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

Offset

0,3

Comments

Without the first two terms, same as A007676 (numerators 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.

Formula

a(n)/A113874(n) -> e.

Maple

a[0]:=1: a[1]:=1: a[2]:=2: for n from 3 to 33 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..33); # Emeric Deutsch, Jan 28 2006

Mathematica

a[0] = a[1] = 1; 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] (* Robert G. Wilson v, Jan 28 2006 *)

Crossrefs

Sequence in context: A041893 A206241 A295333 * A007676 A042443 A042263

Adjacent sequences: A113870 A113871 A113872 * A113874 A113875 A113876

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