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%I #39 Sep 18 2020 06:24:14
%S 1,1,2,3,6,10,20,35,70,126,252,461,922,1702,3404,6315,12630,23494,
%T 46988,87533,175066,326382,652764,1217483,2434966,4542526,9085052,
%U 16950573,33901146,63255670,126511340,236063915,472127830,880983606,1761967212,3287837741
%N Number of paths of length n starting at initial node of the path graph P_11.
%C Also the number of paths along a corridor width 11, starting from one side.
%C 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
%H Johann Cigler, <a href="http://arxiv.org/abs/1501.04750">Some remarks and conjectures related to lattice paths in strips along the x-axis</a>, arXiv:1501.04750 [math.CO], 2015-2016.
%H Nachum Dershowitz, <a href="https://arxiv.org/abs/2006.06516">Between Broadway and the Hudson: A Bijection of Corridor Paths</a>, arXiv:2006.06516 [math.CO], 2020.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,6,0,-9,0,2).
%F 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)).
%F 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
%p X := j -> (-1)^(j/12) - (-1)^(1-j/12):
%p a := k -> add((2 + X(j))*X(j)^k, j in [1, 3, 5, 7, 9, 11])/12:
%p seq(simplify(a(n)), n=0..30); # _Peter Luschny_, Sep 17 2020
%t LinearRecurrence[{0, 6, 0, -9, 0, 2}, {1, 1, 2, 3, 6, 10}, 40] (* _Harvey P. Dale_, Sep 08 2020 *)
%t 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}]]
%t Table[a[n,11], {n,0,40}]//Round (* _Herbert Kociemba_, Sep 14 2020 *)
%o (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
%Y This is row 11 of A094718. Bisections give A087944 (even part), A087946 (odd part).
%Y 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),
%Y A178381 (row 9), A336675 (row 10), this sequence (row 11), A001405 (limit).
%K nonn,easy,walk
%O 0,3
%A _Nachum Dershowitz_, Jul 30 2020
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