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A368436
Irregular triangular array T, read by rows: T(n,k) = number of sums |x-y|+|y-z| = k, where (x,y,z) is a permutation of three distinct numbers x,y,z taken from {0,1,...,n}, for n >= 2, k >= 2.
1
2, 4, 4, 12, 4, 4, 6, 20, 14, 12, 4, 4, 8, 28, 24, 28, 12, 12, 4, 4, 10, 36, 34, 44, 30, 24, 12, 12, 4, 4, 12, 44, 44, 60, 48, 48, 24, 24, 12, 12, 4, 4, 14, 52, 54, 76, 66, 72, 50, 40, 24, 24, 12, 12, 4, 4, 16, 60, 64, 92, 84, 96, 76, 72, 40, 40, 24, 24, 12
OFFSET
1,1
COMMENTS
Row n consists of 2n even positive integers having sum A007531(n+2) = (n+2)!/(n-1)!.
EXAMPLE
Taking n = 2, the permutations of {x,y,z} of {0,1,2} with sums |x-y| + |y-z| = 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 (2,4), representing two 2s and four 3s.
First eight rows:
2 4
4 12 4 4
6 20 14 12 4 4
8 28 24 28 12 12 4 4
10 36 34 44 30 24 12 12 4 4
12 44 44 60 48 48 24 24 12 12 4 4
14 52 54 76 66 72 50 40 24 24 12 12 4 4
16 60 64 92 84 96 76 72 40 40 24 24 12 12 4 4
MATHEMATICA
t[n_] := t[n] = Permutations[-1 + Range[n + 1], {3}];
a[n_, k_] := Select[t[n], Abs[#[[1]] - #[[2]]] + Abs[#[[2]] - #[[3]]] == k &];
u = Table[Length[a[n, k]], {n, 2, 15}, {k, 2, 2 n - 1}];
v = Flatten[u] (* sequence *)
Column[u] (* array *)
CROSSREFS
Cf. A007531, A368434, A368437 (reversed rows).
Sequence in context: A366045 A263382 A286714 * A366616 A186988 A186989
KEYWORD
nonn,tabf
AUTHOR
Clark Kimberling, Dec 25 2023
STATUS
approved