%I A006054 M1396
%S A006054 0,0,1,2,5,11,25,56,126,283,636,1429,3211,7215,16212,36428,81853,183922,
%T A006054 413269,928607,2086561,4688460,10534874,23671647,53189708,119516189,
%U A006054 268550439,603427359,1355888968,3046654856,6845771321,15382308530
%N A006054 a(n)=2a(n-1)+a(n-2)-a(n-3).
%C A006054 Let u(k), v(k), w(k) be be defined by u(1)=1, v(1)=0, w(1)=0 and u(k+1)=u(k)+v(k)+w(k), 
               v(k+1)=u(k)+v(k), w(k+1)=u(k); then {u(n)} = 1,1,3,6,14,31,... (A006356 
               with an extra initial 1), {v(n)} = 0,1,2,5,11,25,... (this sequence 
               with its initial 0 deleted) and {w(n)} = {u(n)} prefixed by an extra 
               0 = A077998 with an extra initial 0. - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Apr 05 2002. Also u(k)^2+v(k)^2+w(k)^2 = u(2k). - Gary Adamson, Dec 
               23 2003.
%C A006054 Form the graph with matrix A=[1, 1, 1; 1, 0, 0; 1, 0, 1]. Then A006054 
               counts walks of length n between the vertex of degree 1 and the vertex 
               of degree 3. - Paul Barry (pbarry(AT)wit.ie), Oct 02 2004
%C A006054 Form the digraph with matrix [1,1,0;1,0,1;1,1,1]. A006054(n) counts walks 
               of length n between the vertices with loops. - Paul Barry (pbarry(AT)wit.ie), 
               Oct 15 2004
%C A006054 a(n), n>1 = round(k*A006356(n-1)), where k = .3568958678... = 1/(1+2*Cos 
               Pi/7) - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 06 2008
%C A006054 A006054(n) = sum of (n-2)-th row terms of triangle A144159. [From Gary 
               W. Adamson (qntmpkt(AT)yahoo.com), Sep 12 2008]
%C A006054 Nonzero terms = INVERT transform of (1, 1, 2, 2, 3, 3,...). Example: 
               56 = (1, 1, 2, 5, 11, 25) dot (3, 3, 2, 2, 1, 1) = (3 + 3 + 4 + 10 
               + 11 + 25). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 20 
               2009]
%D A006054 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A006054 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 A006054 Jay Kappraff, Beyond Measure, A Guided Tour Through Nature, Myth and 
               Number, World Scientific, 2002.
%D A006054 P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), 
               no. 1, 22-31.
%D A006054 Roman Witula, Damian Slota and Adam Warzynski, Quasi-Fibonacci Numbers 
               of the Seventh Order, Journal of Integer Sequences, Vol. 9 (2006), 
               Article 06.4.3.
%H A006054 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A006054 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 A006054 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 A006054 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=434">
               Encyclopedia of Combinatorial Structures 434</a>
%F A006054 G.f.: x^2/(1-2x-x^2+x^3).
%F A006054 Sum_{k, 0<=k<=n+2} a(k) = A077850(n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Sep 07 2006
%F A006054 Let M = the 3 X 3 matrix [1,1,0; 1,2,1; 0,1,2], then M^n*[1,0,0] = [A080937(n-1), 
               A094790(n), A006054(n-1)]. E.g. M^3*[1,0,0] = [5,9,5] = [A080937(2), 
               A094790(3), A006054(2)]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Feb 15 2006
%p A006054 A006054:=z**2/(1-2*z-z**2+z**3); [Conjectured by S. Plouffe in his 1992 
               dissertation.]
%Y A006054 Cf. A006356, A007583, A005578.
%Y A006054 Cf. A080937, A094790.
%Y A006054 A144159 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 12 2008]
%Y A006054 Sequence in context: A151529 A017919 A017920 this_sequence A106805 A094981 
               A097779
%Y A006054 Adjacent sequences: A006051 A006052 A006053 this_sequence A006055 A006056 
               A006057
%K A006054 nonn
%O A006054 0,4
%A A006054 N. J. A. Sloane (njas(AT)research.att.com).