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A052975
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Expansion of (1-2x)(1-x)/(1-5x+6x^2-x^3).
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4
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1, 2, 6, 19, 61, 197, 638, 2069, 6714, 21794, 70755, 229725, 745889, 2421850, 7863641, 25532994, 82904974, 269190547, 874055885, 2838041117, 9215060822, 29921113293, 97153242650, 315454594314, 1024274628963, 3325798821581
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| 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 (kociemba(AT)t-online.de), Jun 11 2004
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), May 29 2010: (Start)
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.
(End)
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LINKS
| INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1047
Index to sequences with linear recurrences with constant coefficients, signature (5,-6,1).
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FORMULA
| G.f.: -(-1+2*x)*(-1+x)/(-1+5*x-6*x^2+x^3)
a(n)=A028495(2n). - Floor van Lamoen (fvlamoen(AT)hotmail.com), Nov 02 2005
Sum(1/7*(2-3*_alpha+_alpha^2)*_alpha^(-1-n), _alpha=RootOf(-1+5*_Z-6*_Z^2+_Z^3))
a(n)=(2/7)*Sum(r, 1, 6, Sin(r*3*Pi/7)^2(2Cos(r*Pi/7))^(2n)); a(n) = 5a(n-1)-6a(n-2)+a(n-3). - Herbert Kociemba (kociemba(AT)t-online.de), Jun 11 2004
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MAPLE
| spec := [S, {S=Sequence(Prod(Union(Sequence(Prod(Sequence(Z), Z)), Sequence(Z)), Z))}, unlabeled ]: seq(combstruct[count ](spec, size=n), n=0..20);
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), May 29 2010: (Start)
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);
(End)
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CROSSREFS
| Cf. A060557.
Cf. A028495, A078038 and A094790.
Sequence in context: A014010 A022015 A138747 * A035929 A071646 A114627
Adjacent sequences: A052972 A052973 A052974 * A052976 A052977 A052978
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KEYWORD
| easy,nonn
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AUTHOR
| encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jun 06 2000
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