A286764 - OEIS

%I #10 Jun 28 2022 03:23:39

%S 1,3,16,109,855,7298,65838,617118,5946781,58506642,584894463,

%T 5921596628,60565217546,624644829720,6487216108058,67767838847144,

%U 711463437534474,7501409431304796,79386836213817417,842882477863610604,8974911258934880498,95806877080558096428

%N Total number of nodes summed over all lattice paths from (0,0) to (n,n) that do not go below the x-axis or above the diagonal x=y and consist of steps D=(1,-1), H=(1,0) and S=(0,1).

%H Alois P. Heinz, <a href="/A286764/b286764.txt">Table of n, a(n) for n = 0..960</a>

%F a(n) ~ c * phi^(5*n) / sqrt(n), where phi = A001622 is the golden ratio and c = 0.036755631845424682385214848270310481743236419858524834059514156934711202... - _Vaclav Kotesovec_, Sep 11 2021

%p b:= proc(x, y) option remember; `if`(y<0 or y>x, 0, `if`(x=0, [1$2],

%p       (p-> p+[0, p[1]])(b(x-1, y)+b(x, y-1)+b(x-1, y+1))))

%p     end:

%p a:= n-> b(n$2)[2]:

%p seq(a(n), n=0..30);

%t b[x_, y_] := b[x, y] = If[y < 0 || y > x, 0, If[x == 0, {1, 1}, Function[

%t      p, p + {0, p[[1]]}][b[x - 1, y] + b[x, y - 1] + b[x - 1, y + 1]]]];

%t a[n_] := b[n, n][[2]];

%t Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Jun 28 2022, after _Alois P. Heinz_ *)

%Y Cf. A001622, A224776.

%K nonn

%O 0,2

%A _Alois P. Heinz_, May 14 2017