This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A336675 #40 Sep 18 2020 06:26:13
%S A336675 1,1,2,3,6,10,20,35,70,126,251,460,911,1690,3327,6225,12190,22950,
%T A336675 44744,84626,164407,312019,604487,1150208,2223504,4239225,8181175,
%U A336675 15621426,30108147,57556155,110820165,212037241,407946421,781074572,1501844193,2877011660,5529362694
%N A336675 Number of paths of length n starting at initial node of the path graph P_10.
%C A336675 Also the number of paths along a corridor width 10, starting from one side.
%C A336675 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 A336675 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 A336675 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 A336675 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,-3,-3,1).
%F A336675 From _Stefano Spezia_, Jul 30 2020: (Start)
%F A336675 G.f.: (1 - 3*x^2 + x^4)/(1 - x - 4*x^2 + 3*x^3 + 3*x^4 - x^5).
%F A336675 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 A336675 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 A336675 X := j -> (-1)^(j/11) - (-1)^(1-j/11):
%p A336675 a := k -> add((2 + X(j))*X(j)^k, j in [1, 3, 5, 7, 9])/11:
%p A336675 seq(simplify(a(n)), n=0..30); # _Peter Luschny_, Sep 17 2020
%t A336675 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 A336675 Table[a[n,10],{n,0,40}]//Round (* _Herbert Kociemba_, Sep 14 2020 *)
%o A336675 (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 A336675 This is row 10 of A094718. Bisections give A224514 (even part), A216710 (odd part).
%Y A336675 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 A336675 nonn,easy,walk
%O A336675 0,3
%A A336675 _Nachum Dershowitz_, Jul 30 2020