%I A006053 M2358
%S A006053 0,0,1,1,3,4,9,14,28,47,89,155,286,507,924,1652,2993,5373,9707,17460,
%T A006053 31501,56714,102256,184183,331981,598091,1077870,1942071,3499720,
%U A006053 6305992,11363361,20475625,36896355,66484244,119801329,215873462
%N A006053 a(n)=a(n-1)+2a(n-2)-a(n-3).
%C A006053 a(n+1)=S(n) for n>=1, where S(n) is the number of 01-words of length 
               n, having first letter 1, in which all runlengths of 1's are odd. 
               Example: S(4) counts 1000,1001,1010,1110. See A077865. - Clark Kimberling 
               (ck6(AT)evansville.edu), Jun 26 2004
%C A006053 Counts walks of length n between the first and second nodes of P_3, to 
               which a loop has been added at the end. Let A be the adjacency matrix 
               of the graph P_3 with a loop added at the end. A is a 'reverse Jordan 
               matrix' [0,0,1;0,1,1;1,1,0]. a(n) is obtained by taking the (1,2) 
               element of A^n. - Paul Barry (pbarry(AT)wit.ie), Jul 16 2004
%C A006053 Interleaves A094790 and A094789. - Paul Barry (pbarry(AT)wit.ie), Oct 
               30 2004
%D A006053 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A006053 S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques 
               Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%D A006053 R. Chapman, Eigenvalues of a Bidiagonal Matrix, Amer. Math. Monthly, 
               111 (2004) p. 441
%H A006053 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 A006053 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 A006053 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=433">
               Encyclopedia of Combinatorial Structures 433</a>
%F A006053 a(n+2)=A094790(n/2+1)(1+(-1)^n)/2+A094789((n+1)/2)(1-(-1)^n)/2 - Paul 
               Barry (pbarry(AT)wit.ie), Oct 30 2004
%F A006053 First differences of A028495. - Floor van Lamoen (fvlamoen(AT)hotmail.com), 
               Nov 02 2005
%F A006053 G.f.=x^2/(1-x-2x^2+x^3). - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Dec 14 2004
%p A006053 a[0]:=0: a[1]:=0: a[2]:=1: for n from 3 to 40 do a[n]:=a[n-1]+2*a[n-2]-a[n-3] 
               od:seq(a[n],n=0..40); (Deutsch)
%p A006053 A006053:=z**2/(1-z-2*z**2+z**3); [Conjectured by S. Plouffe in his 1992 
               dissertation.]
%Y A006053 Cf. A096975, A096976.
%Y A006053 Sequence in context: A014596 A002823 A109509 this_sequence A051841 A096081 
               A054162
%Y A006053 Adjacent sequences: A006050 A006051 A006052 this_sequence A006054 A006055 
               A006056
%K A006053 nonn,easy
%O A006053 0,5
%A A006053 N. J. A. Sloane (njas(AT)research.att.com).
%E A006053 More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 14 2004