A336675 Number of paths of length n starting at initial node of the path graph P_10.

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

Offset

0,3

Comments

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

Links

Table of n, a(n) for n=0..36.

Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750 [math.CO], 2015-2016.

Nachum Dershowitz, Between Broadway and the Hudson: A Bijection of Corridor Paths, arXiv:2006.06516 [math.CO], 2020.

Index entries for linear recurrences with constant coefficients, signature (1,4,-3,-3,1).

Formula

From Stefano Spezia, Jul 30 2020: (Start)

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

Maple

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:
seq(simplify(a(n)), n=0..30); # Peter Luschny, Sep 17 2020

Mathematica

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}]]
Table[a[n, 10], {n, 0, 40}]//Round (* Herbert Kociemba, Sep 14 2020 *)

Prog

(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

Crossrefs

This is row 10 of A094718. Bisections give A224514 (even part), A216710 (odd part).

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).

Sequence in context: A026034 A178381 A037031 * A336678 A056202 A001405

Adjacent sequences: A336672 A336673 A336674 * A336676 A336677 A336678

Keyword

nonn,easy,walk

Author

Nachum Dershowitz, Jul 30 2020

Status

approved