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A336678
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Number of paths of length n starting at initial node of the path graph P_11.
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3
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1, 1, 2, 3, 6, 10, 20, 35, 70, 126, 252, 461, 922, 1702, 3404, 6315, 12630, 23494, 46988, 87533, 175066, 326382, 652764, 1217483, 2434966, 4542526, 9085052, 16950573, 33901146, 63255670, 126511340, 236063915, 472127830, 880983606, 1761967212, 3287837741
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OFFSET
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0,3
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COMMENTS
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Also the number of paths along a corridor width 11, starting from one side.
In general, a(n,m) = (2^n/(m+1))*Sum_{r=1..m} (1-(-1)^r)*cos(Pi*r/(m+1))^n*(1+cos(Pi*r/(m+1))) gives the number of paths of length n starting at the initial node on the path graph P_m. Here we have m=11. - Herbert Kociemba, Sep 14 2020
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LINKS
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FORMULA
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G.f.: -(x^5+3*x^4-3*x^3-4*x^2+x+1)/((2*x^2-1)*(x^4-4*x^2+1)).
a(n) = (2^n/12)*Sum_{r=1..11} (1-(-1)^r)*cos(Pi*r/12)^n*(1+cos(Pi*r/12)). - Herbert Kociemba, Sep 14 2020
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MAPLE
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X := j -> (-1)^(j/12) - (-1)^(1-j/12):
a := k -> add((2 + X(j))*X(j)^k, j in [1, 3, 5, 7, 9, 11])/12:
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MATHEMATICA
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LinearRecurrence[{0, 6, 0, -9, 0, 2}, {1, 1, 2, 3, 6, 10}, 40] (* Harvey P. Dale, Sep 08 2020 *)
a[n_, m_]:=2^(n+1)/(m+1) Module[{x=(Pi r)/(m+1)}, Sum[Cos[x]^n (1+Cos[x]), {r, 1, m, 2}]]
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PROG
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(PARI) my(x='x+O('x^44)); Vec(-(x^5+3*x^4-3*x^3-4*x^2+x+1)/((2*x^2-1)*(x^4-4*x^2+1))) \\ Joerg Arndt, Jul 31 2020
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CROSSREFS
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KEYWORD
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nonn,easy,walk
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AUTHOR
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STATUS
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approved
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