%I A000213 M2454 N0975
%S A000213 1,1,1,3,5,9,17,31,57,105,193,355,653,1201,2209,4063,7473,13745,25281,
%T A000213 46499,85525,157305,289329,532159,978793,1800281,3311233,6090307,
%U A000213 11201821,20603361,37895489,69700671,128199521,235795681,433695873
%N A000213 Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) with a(0)=a(1)=a(2)=1.
%C A000213 Number of (n-1)-bit binary sequences with each one adjacent to a zero. 
               - Ron Hardin (rhhardin(AT)att.net), Dec 24 2007
%C A000213 The binomial transform is A099216. The inverse binomial transform is 
               (-1)^n*A124395(n). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Aug 19 2008]
%C A000213 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 27 2009: 
               (Start)
%C A000213 Equals INVERT transform of (1, 0, 2, 0, 2, 0, 2,...). a(6) = 17 =
%C A000213 (1, 1, 1, 3, 5, 9) dot (0, 2, 0, 2, 0, 1) = (0 + 2 + 0 + 6 + 0 + 9) = 
               17. (End)
%D A000213 B. G. Baumgart, Letter to the editor, Fib. Quart. 2 (1964), 260, 302.
%D A000213 M. Feinberg, Fibonacci-Tribonacci, Fib. Quart. 1(#3) (1963), 71-74.
%D A000213 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000213 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A000213 T. D. Noe, <a href="b000213.txt">Table of n, a(n) for n = 0..200</a>
%H A000213 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Fxtbook</a>
%H A000213 Nick Hobson, <a href="a000213.py.txt">Python program for this sequence</
               a>
%H A000213 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A000213 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A000213 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               TribonacciNumber.html">Tribonacci Number</a>
%H A000213 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%F A000213 G.f.: (1-x)*(1+x)/(1-x-x^2-x^3). - Ralf Stephan, Feb 11 2004
%F A000213 a(n) = rightmost term of M^n * [1 1 1], where M = the 3X3 matrix [1 1 
               1 / 1 0 0 / 0 1 0]. (M^n * [1 1 1]= [a(n+2) a(n+1) a(n)]). a(n)/a(n-1) 
               tends to the tribonacci constant, 1.839286755...; an eigenvalue of 
               M and a root of x^3 - x^2 - x - 1 = 0. - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Dec 17 2004
%F A000213 a(n)=A001590(n+3)-A001590(n+2); a(n+1)-a(n)=2*A000073(n); a(n)=A000073(n+3)-A000073(n+1). 
               - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), May 22 2006
%F A000213 a(n)=A001590(n)+A001590(n+1) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Sep 25 2006
%F A000213 a(n) ~ (F - 1) * T^n, where F = A086254 and T = A058265. [From Charles 
               R Greathouse IV Nov 09 2008]
%p A000213 K:=(1-z^2)/(1-z-z^2-z^3): Kser:=series(K, z=0, 45): seq((coeff(Kser, 
               z, n)), n= 0..34); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Nov 08 2007
%p A000213 A000213:=(z-1)*(1+z)/(-1+z+z**2+z**3); [S. Plouffe in his 1992 dissertation.]
%t A000213 a=1;b=1;c=1;lst={a, b, c};Do[d=a+b+c;AppendTo[lst, d];a=b;b=c;c=d, {n, 
               5!}];lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Sep 30 
               2008]
%o A000213 sage: from sage.combinat.sloane_functions import recur_gen3 sage: it 
               = recur_gen3(1,1,1,1,1,1) sage: [it.next() for i in range(35)] - 
               Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008
%Y A000213 Cf. A000288, A000322, A000383, A046735, A060455.
%Y A000213 Sequence in context: A102475 A066173 A114322 this_sequence A074858 A074860 
               A135728
%Y A000213 Adjacent sequences: A000210 A000211 A000212 this_sequence A000214 A000215 
               A000216
%K A000213 easy,nonn,nice
%O A000213 0,4
%A A000213 N. J. A. Sloane (njas(AT)research.att.com).