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A330617 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. 1
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, 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, 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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,6
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
E. Krom and M. M. Roughan, Path Counting and Eulerian Numbers, Girls' Angle Bulletin, Vol. 13, No. 3 (2020), 8-10.
FORMULA
For k odd: T(n, k) = 2^((k-1)/2)*(ceiling(n/2) - (k-1)/2).
For k even: T(n, k) = 2^(k/2).
T(2*n-1, 2*k-1) = A130128(n, k).
EXAMPLE
First few rows of the triangle are:
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, 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, 16, 32;
...
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}.
MATHEMATICA
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 *)
PROG
(PARI) T(n, k)={if(k%2, 2^(k\2)*((n+1)\2 - k\2), 2^(k/2))} \\ Andrew Howroyd, Mar 17 2020
CROSSREFS
Cf. A130128.
Sequence in context: A235708 A353929 A297770 * A343240 A145866 A103318
KEYWORD
nonn,easy,tabl,walk
AUTHOR
Grace Work, Mar 01 2020
STATUS
approved

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Last modified February 21 10:54 EST 2024. Contains 370233 sequences. (Running on oeis4.)