OFFSET
0,3
COMMENTS
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
LINKS
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 (0,6,0,-9,0,2).
FORMULA
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
MAPLE
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:
seq(simplify(a(n)), n=0..30); # Peter Luschny, Sep 17 2020
MATHEMATICA
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}]]
Table[a[n, 11], {n, 0, 40}]//Round (* Herbert Kociemba, Sep 14 2020 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,easy,walk
AUTHOR
Nachum Dershowitz, Jul 30 2020
STATUS
approved