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%I #62 Feb 12 2022 17:55:37
%S 1,2,6,19,61,197,638,2069,6714,21794,70755,229725,745889,2421850,
%T 7863641,25532994,82904974,269190547,874055885,2838041117,9215060822,
%U 29921113293,97153242650,315454594314,1024274628963,3325798821581,10798800928441,35063486341682
%N Expansion of (1-2*x)*(1-x)/(1-5*x+6*x^2-x^3).
%C Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 7 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = 3, s(2n) = 3. - _Herbert Kociemba_, Jun 11 2004
%C Counts all paths of length (2*n), n>=0, starting at the initial node and ending on the nodes 1, 2, 3, 4 and 5 on the path graph P_6, see the second Maple program. - _Johannes W. Meijer_, May 29 2010
%H G. C. Greubel, <a href="/A052975/b052975.txt">Table of n, a(n) for n = 0..1000</a>
%H Paul Barry, <a href="https://arxiv.org/abs/2104.01644">Centered polygon numbers, heptagons and nonagons, and the Robbins numbers</a>, arXiv:2104.01644 [math.CO], 2021.
%H Nachum Dershowitz, <a href="https://arxiv.org/abs/2006.06516">Between Broadway and the Hudson: A Bijection of Corridor Paths</a>, arXiv:2006.06516 [math.CO], 2020.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=1047">Encyclopedia of Combinatorial Structures 1047</a>
%H László Németh and László Szalay, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Nemeth/nemeth8.html">Sequences Involving Square Zig-Zag Shapes</a>, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
%H Roman Witula, Damian Slota and Adam Warzynski, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Slota/slota57.html">Quasi-Fibonacci Numbers of the Seventh Order</a>, J. Integer Seq., 9 (2006), Article 06.4.3.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-6,1).
%F G.f.: (1-2*x)*(1-x)/(1-5*x+6*x^2-x^3).
%F a(n) = A028495(2*n). - _Floor van Lamoen_, Nov 02 2005
%F a(n) = Sum (1/7*(2-3*_alpha+_alpha^2)*_alpha^(-1-n), _alpha=RootOf(-1+5*_Z-6*_Z^2+_Z^3))
%F From _Herbert Kociemba_, Jun 11 2004: (Start)
%F a(n) = (2/7)*Sum_{r=1..6} sin(r*3*Pi/7)^2*(2*cos(r*Pi/7))^(2*n).
%F a(n) = 5*a(n-1) - 6*a(n-2) + a(n-3). (End)
%F a(n) = 2^n*A(n;1/2) = (1/7)*(s(2)^2*c(4)^(2n) + s(4)^2*c(1)^(2n) + s(1)^2*c(2)^(2n)), where c(j):=2*cos(2Pi*j/7) and s(j):=2*sin(2*Pi*j/7). Here A(n;d), n in N, d in C denotes the respective quasi-Fibonacci number - see A121449 and Witula-Slota-Warzynski paper for details (see also A094789, A085810, A077998, A006054, A121442). I note that my and the respective Herbert Kociemba's formulas are "compatible". - _Roman Witula_, Aug 09 2012
%F a(n) = A005021(n)-3*A005021(n-1)+2*A005021(n-2). - _R. J. Mathar_, Feb 27 2019
%p spec := [S,{S=Sequence(Prod(Union(Sequence(Prod(Sequence(Z),Z)),Sequence(Z)),Z))},unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
%p with(GraphTheory):G:=PathGraph(6): A:= AdjacencyMatrix(G): nmax:=25; n2:=2*nmax+1: for n from 0 to n2 do B(n):=A^n; a(n):=add(B(n)[k,1],k=1..5); od: seq(a(2*n),n=0..nmax); # _Johannes W. Meijer_, May 29 2010
%t LinearRecurrence[{5,-6,1}, {1,2,6}, 50] (* _Roman Witula_, Aug 09 2012 *)
%t CoefficientList[Series[(1 - 2 x) (1 - x)/(1 - 5 x + 6 x^2 - x^3), {x, 0, 40}], x] (* _Vincenzo Librandi_, Sep 18 2015 *)
%o (Magma) I:=[1,2,6]; [n le 3 select I[n] else 5*Self(n-1)-6*Self(n-2)+Self(n-3): n in [1..30]]; // _Vincenzo Librandi_, Sep 18 2015
%o (PARI) x='x+O('x^30); Vec((1-2*x)*(1-x)/(1-5*x+6*x^2-x^3)) \\ _G. C. Greubel_, Apr 19 2018
%Y Cf. A060557.
%Y Cf. A028495, A078038 and A094790.
%K easy,nonn
%O 0,2
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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