OFFSET
0,5
COMMENTS
In a convex n-gon, the number of cycles using k non-repeated vertices and fewer than 3 vertices (2 sides) in a row.
FORMULA
T(n,k) = n*(5*A340106(n-1,k-1) - 2*Z(n,k) - Z(n-1,k-1) - 2*S(n,k) - 2*S(n-2,k-2)) except for T(n,0)=1, where S(n,k) = 2*A340106(n-1,k-1) - 2*A340106(n-2,k-2) + S(n-3,k-3), S(n,k)=0 for k <= 0, Z(n,k) = 2*A340106(n-1,k-1) - S(n,k) - V(n-1,k-1), Z(n,k)=0 for k <= 0, V(n,k) = Z(n-1,k-1) - V(n-1,k-1), V(n,k)=0 for k <= 0 except for V(2,2)=2.
EXAMPLE
n\k 0 1 2 3 4 5 6 7 8
0 1
1 1 1
2 1 2 2
3 1 3 6 0
4 1 4 12 0 16
5 1 5 20 30 80 60
6 1 6 30 84 264 480 456
7 1 7 42 168 672 1890 3612 3458
8 1 8 56 288 1424 5440 15744 30352 29296
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Xiangyu Chen, Dec 28 2020
EXTENSIONS
Terms of column 2 corrected by Xiangyu Chen, Aug 19 2022
STATUS
approved