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A340108
Triangle read by rows: T(n,k) is the number of permutations of k elements from [1..n] in a circle with longest consecutive chain size less than 3, when 1 and n are considered to be consecutive, and rotations are considered to be distinct.
2
1, 1, 1, 1, 2, 2, 1, 3, 6, 0, 1, 4, 12, 0, 16, 1, 5, 20, 30, 80, 60, 1, 6, 30, 84, 264, 480, 456, 1, 7, 42, 168, 672, 1890, 3612, 3458, 1, 8, 56, 288, 1424, 5440, 15744, 30352, 29296, 1, 9, 72, 450, 2664, 12870, 50004, 145656, 283104, 275166, 1, 10, 90, 660, 4560, 26640, 130080, 508060, 1488960, 2909700, 2843980
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