A113873 - OEIS

%I #24 Jun 04 2019 10:37:17

%S 1,1,2,3,8,11,19,87,106,193,1264,1457,2721,23225,25946,49171,517656,

%T 566827,1084483,13580623,14665106,28245729,410105312,438351041,

%U 848456353,14013652689,14862109042,28875761731,534625820200

%N 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).

%C Without the first two terms, same as A007676 (numerators of convergents to e). - _Jonathan Sondow_, Aug 16 2006

%H T. D. Noe, <a href="/A113873/b113873.txt">Table of n, a(n) for n = 0..201</a>

%H H. Cohn, <a href="https://www.jstor.org/stable/27641837">A short proof of the simple continued fraction expansion of e</a>, Amer. Math. Monthly, 113 (No. 1, 2006), 57-62.

%H J. Sondow, <a href="https://www.jstor.org/stable/27642006">A geometric proof that e is irrational and a new measure of its irrationality</a>, Amer. Math. Monthly 113 (2006) 637-641.

%H J. Sondow, <a href="https://arxiv.org/abs/0704.1282">A geometric proof that e is irrational and a new measure of its irrationality</a>, arXiv:0704.1282 [math.HO], 2007-2010.

%H J. Sondow and K. Schalm, <a href="http://arxiv.org/abs/0709.0671">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</a>, Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010.

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

%p 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

%t 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 *)

%K easy,nonn

%O 0,3

%A _N. J. A. Sloane_, Jan 27 2006

%E More terms from _Robert G. Wilson v_ and _Emeric Deutsch_, Jan 28 2006