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A094790
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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) = 1, s(2n) = 3.
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10
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1, 3, 9, 28, 89, 286, 924, 2993, 9707, 31501, 102256, 331981, 1077870, 3499720, 11363361, 36896355, 119801329, 388991876, 1263047761, 4101088878, 13316149700, 43237262993, 140390505643, 455845099957, 1480119728920
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| In general a(n)= 2/m*Sum_{r=1..m-1} Sin(r*j*Pi/m)Sin(r*k*Pi/m)(2Cos(r*Pi/m))^(2n)) counts (s(0), s(1), ..., s(2n)) such that 0 < s(i) < m and |s(i) - s(i-1)| = 1 for i = 1,2,....,2n, s(0) = j, s(2n) = k.
With interpolated zeros (0,0,1,0,3,0,9...), counts walks of length n between the first and third nodes of P_6. - Paul Barry (pbarry(AT)wit.ie), Jan 26 2005
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), May 29 2010: (Start)
Counts all paths of length (2*n+1), 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 Maple program.
(End)
With offset 0 = the INVERT transform of A055588 - Gary W. Adamson (qntmpkt(AT)yahoo.com), APR 1 2011.
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FORMULA
| a(n)=2/7*Sum_{k=1..6} Sin(Pi*k/7)Sin(3Pi*k/7)(2Cos(Pi*k/7))^(2n); a(n) = 5a(n-1)-6a(n-2)+a(n-3); G.f.: x(-1+2x)/(-1+5x-6x^2+x^3)
a(n) = right-most term in M^n * [1,0,0] where M = the 3 X 3 matrix [2,1,1; 1,2,0; 1,0,1]. E.g. M^3 * [1,0,0] = [19,14,9]. right term = 9 = a(3). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 04 2006
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MAPLE
| with(GraphTheory):G:=PathGraph(6): A:= AdjacencyMatrix(G): nmax:=24; 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+1), n=0..nmax); [From Johannes W. Meijer (meijgia(AT)hotmail.com), May 29 2010]
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MATHEMATICA
| f[n_] := FullSimplify[ TrigToExp[(2/7)Sum[ Sin[Pi*k/7]Sin[3Pi*k/7](2Cos[Pi*k/7])^(2n), {k, 1, 6}]]]; Table[ f[n], {n, 25}] (from Robert G. Wilson v Jun 18 2004)
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CROSSREFS
| Cf. A080937, A094789, A005021.
Cf. A028495, A052975 and A078038.
Cf. A055588.
Sequence in context: A170953 A199104 A049220 * A007822 A094164 A094803
Adjacent sequences: A094787 A094788 A094789 * A094791 A094792 A094793
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KEYWORD
| nonn
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AUTHOR
| Herbert Kociemba (kociemba(AT)t-online.de), Jun 11 2004
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