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A368437
Irregular triangular array T, read by rows: T(n,k) = number of sums |x-y| + |y-z| = 2n+1-k, where (x,y,z) is a permutation of three distinct numbers taken from {0,1,...,n}, for n >= 2, k >= 2.
14
4, 2, 4, 4, 12, 4, 4, 4, 12, 14, 20, 6, 4, 4, 12, 12, 28, 24, 28, 8, 4, 4, 12, 12, 24, 30, 44, 34, 36, 10, 4, 4, 12, 12, 24, 24, 48, 48, 60, 44, 44, 12, 4, 4, 12, 12, 24, 24, 40, 50, 72, 66, 76, 54, 52, 14, 4, 4, 12, 12, 24, 24, 40, 40, 72, 76, 96, 84, 92
OFFSET
1,1
COMMENTS
Row n consists of 2n even positive integers having sum A007531(n+2) = (n+2)!/(n-1)!. The limiting row, (4, 4, 12, 12, 24, 24, 40, 40, ...) consists of repeated terms of (A046092(n+1)) = (4, 12, 24, 40, ...).
EXAMPLE
Taking n = 2, the permutations of {x,y,z} of {0,1,2} with sums |x-y| + |y-z| = 2n+1-k, for k = 2,3, are as follows:
012: |0-1| + |1-2| = 2
021: |0-2| + |2-1| = 3
102: |1-0| + |0-2| = 3
120: |1-2| + |2-0| = 3
201: |2-0| + |0-1| = 3
210: |2-1| + |1-0| = 2
so that row 1 of the array is (4,2), representing four 2s and two 3s.
First eight rows:
4 2
4 4 12 4
4 4 12 14 20 6
4 4 12 12 28 24 28 8
4 4 12 12 24 30 44 34 36 10
4 4 12 12 24 24 48 48 60 44 44 12
4 4 12 12 24 24 40 50 72 66 76 54 52 14
4 4 12 12 24 24 40 40 72 76 96 84 92 64 60 16
MATHEMATICA
t[n_] := t[n] = Permutations[-1 + Range[n + 1], {3}];
a[n_, k_] := Select[t[n], Abs[#[[1]] - #[[2]]] + Abs[#[[2]] - #[[3]]] == 2n+1-k &];
u = Table[Length[a[n, k]], {n, 2, 15}, {k, 2, 2 n - 1}];
v = Flatten[u] (* sequence *)
Column[u] (* array *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Clark Kimberling, Dec 25 2023
STATUS
approved