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A336675
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Number of paths of length n starting at initial node of the path graph P_10.
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3
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1, 1, 2, 3, 6, 10, 20, 35, 70, 126, 251, 460, 911, 1690, 3327, 6225, 12190, 22950, 44744, 84626, 164407, 312019, 604487, 1150208, 2223504, 4239225, 8181175, 15621426, 30108147, 57556155, 110820165, 212037241, 407946421, 781074572, 1501844193, 2877011660, 5529362694
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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 10, 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=10. - Herbert Kociemba, Sep 14 2020
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LINKS
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FORMULA
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G.f.: (1 - 3*x^2 + x^4)/(1 - x - 4*x^2 + 3*x^3 + 3*x^4 - x^5).
a(n) = a(n-1) + 4*a(n-2) - 3*a(n-3) - 3*a(n-4) + a(n-5) for n > 4. (End)
a(n) = (2^n/11)*Sum_{r=1..10} (1-(-1)^r)*cos(Pi*r/11)^n*(1+cos(Pi*r/11)). - Herbert Kociemba, Sep 14 2020
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MAPLE
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X := j -> (-1)^(j/11) - (-1)^(1-j/11):
a := k -> add((2 + X(j))*X(j)^k, j in [1, 3, 5, 7, 9])/11:
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MATHEMATICA
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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((1 - 3*x^2 + x^4)/(1 - x - 4*x^2 + 3*x^3 + 3*x^4 - x^5)) \\ Joerg Arndt, Jul 31 2020
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CROSSREFS
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Cf. A000004 (row 0), A000007 (row 1), A000012 (row 2), A016116 (row 3), A000045 (row 4), A038754 (row 5), A028495 (row 6), A030436 (row 7), A061551 (row 8), A178381 (row 9), this sequence (row 10), A336678 (row 11), A001405 (limit).
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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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