cp's OEIS Frontend

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.

%I A340108 #19 Sep 11 2022 09:29:39
%S A340108 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,
%T A340108 456,1,7,42,168,672,1890,3612,3458,1,8,56,288,1424,5440,15744,30352,
%U A340108 29296,1,9,72,450,2664,12870,50004,145656,283104,275166,1,10,90,660,4560,26640,130080,508060,1488960,2909700,2843980
%N 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.
%C A340108 In a convex n-gon, the number of cycles using k non-repeated vertices and fewer than 3 vertices (2 sides) in a row.
%F A340108 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.
%e A340108 n\k   0     1      2      3     4     5     6     7     8
%e A340108 0     1
%e A340108 1     1     1
%e A340108 2     1     2      2
%e A340108 3     1     3      6      0
%e A340108 4     1     4     12      0     16
%e A340108 5     1     5     20     30     80    60
%e A340108 6     1     6     30     84    264   480    456
%e A340108 7     1     7     42    168    672  1890   3612  3458
%e A340108 8     1     8     56    288   1424  5440  15744 30352 29296
%Y A340108 Cf. A338526, A338838, A338849.
%Y A340108 Cf. A340106, A340107.
%K A340108 nonn,tabl
%O A340108 0,5
%A A340108 _Xiangyu Chen_, Dec 28 2020
%E A340108 Terms of column 2 corrected by _Xiangyu Chen_, Aug 19 2022