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%I #40 Sep 18 2020 06:26:13
%S 1,1,2,3,6,10,20,35,70,126,251,460,911,1690,3327,6225,12190,22950,
%T 44744,84626,164407,312019,604487,1150208,2223504,4239225,8181175,
%U 15621426,30108147,57556155,110820165,212037241,407946421,781074572,1501844193,2877011660,5529362694
%N Number of paths of length n starting at initial node of the path graph P_10.
%C Also the number of paths along a corridor width 10, 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=10. - _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_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,-3,-3,1).
%F From _Stefano Spezia_, Jul 30 2020: (Start)
%F G.f.: (1 - 3*x^2 + x^4)/(1 - x - 4*x^2 + 3*x^3 + 3*x^4 - x^5).
%F a(n) = a(n-1) + 4*a(n-2) - 3*a(n-3) - 3*a(n-4) + a(n-5) for n > 4. (End)
%F 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
%p X := j -> (-1)^(j/11) - (-1)^(1-j/11):
%p a := k -> add((2 + X(j))*X(j)^k, j in [1, 3, 5, 7, 9])/11:
%p seq(simplify(a(n)), n=0..30); # _Peter Luschny_, Sep 17 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,10],{n,0,40}]//Round (* _Herbert Kociemba_, Sep 14 2020 *)
%o (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
%Y This is row 10 of A094718. Bisections give A224514 (even part), A216710 (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), A178381 (row 9), this sequence (row 10), A336678 (row 11), A001405 (limit).
%K nonn,easy,walk
%O 0,3
%A _Nachum Dershowitz_, Jul 30 2020
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