A330617 - OEIS

%I #43 Jan 07 2024 16:31:29

%S 1,1,1,1,1,2,1,2,2,2,1,2,2,2,4,1,3,2,4,4,4,1,3,2,4,4,4,8,1,4,2,6,4,8,

%T 8,8,1,4,2,6,4,8,8,8,16,1,5,2,8,4,12,8,16,16,16,1,5,2,8,4,12,8,16,16,

%U 16,32,1,6,2,10,4,16,8,24,16,32,32,32,1,6,2,10,4,16,8,24,16,32,32,32,64

%N Triangle read by rows: T(n,k) is the number of paths from node 0 to k in a directed graph with n+1 vertices labeled 0, 1, ..., n and edges leading from i to i+1 for all i, and from i to i+2 for even i and from i to i-2 for odd i.

%H Andrew Howroyd, <a href="/A330617/b330617.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)

%H E. Krom and M. M. Roughan, <a href="http://girlsangle.org/page/bulletin-archive/GABv13n03E.pdf">Path Counting and Eulerian Numbers</a>, Girls' Angle Bulletin, Vol. 13, No. 3 (2020), 8-10.

%F For k odd: T(n, k) = 2^((k-1)/2)*(ceiling(n/2) - (k-1)/2).

%F For k even: T(n, k) = 2^(k/2).

%F T(2*n-1, 2*k-1) = A130128(n, k).

%e First few rows of the triangle are:

%e   1;

%e   1,  1;

%e   1,  1,  2;

%e   1,  2,  2,  2;

%e   1,  2,  2,  2,  4;

%e   1,  3,  2,  4,  4,  4;

%e   1,  3,  2,  4,  4,  4,  8;

%e   1,  4,  2,  6,  4,  8,  8,  8;

%e   1,  4,  2,  6,  4,  8,  8,  8, 16;

%e   1,  5,  2,  8,  4, 12,  8, 16, 16, 16;

%e   1,  5,  2,  8,  4, 12,  8, 16, 16, 16, 32;

%e   ...

%e For n=6 and k=3, T(6,3)=4 is the number of paths from node 0 to node 3 along the directed network: {0,1,2,3}, {0,2,3}, {0,2,4,5,3}, {0,1,2,4,5,3}.

%t Table[If[EvenQ@ k, 2^(k/2), 2^((k - 1)/2)*(Ceiling[n/2] - (k - 1)/2)], {n, 0, 12}, {k, 0, n}] // Flatten (* _Michael De Vlieger_, Mar 23 2020 *)

%o (PARI) T(n,k)={if(k%2, 2^(k\2)*((n+1)\2 - k\2), 2^(k/2))} \\ _Andrew Howroyd_, Mar 17 2020

%Y Cf. A130128.

%K nonn,easy,tabl,walk

%O 0,6

%A _Grace Work_, Mar 01 2020