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A368604
Triangular array T, read by rows: T(n,k) = number of sums |x-y|+|y-z| = k, where x,y,z are in {1,2,...,n} and x < y and y > z.
5
0, 1, 0, 2, 2, 1, 3, 4, 4, 2, 4, 6, 7, 6, 4, 5, 8, 10, 10, 9, 6, 6, 10, 13, 14, 14, 12, 9, 7, 12, 16, 18, 19, 18, 16, 12, 8, 14, 19, 22, 24, 24, 23, 20, 16, 9, 16, 22, 26, 29, 30, 30, 28, 25, 20, 10, 18, 25, 30, 34, 36, 37, 36, 34, 30, 25, 11, 20, 28, 34, 39
OFFSET
1,4
COMMENTS
The difference sequence of the k-th column is (k), for k >= 1.
EXAMPLE
First nine rows:
0
1 0
2 2 1
3 4 4 2
4 6 7 6 4
5 8 10 10 9 6
6 10 13 14 14 12 9
7 12 16 18 19 18 16 12
8 14 19 22 24 24 23 20 16
For n=3, there are 5 triples (x,y,z) having x < y and y > z:
121: |x-y| + |y-z| = 2
131: |x-y| + |y-z| = 4
132: |x-y| + |y-z| = 3
231: |x-y| + |y-z| = 3
232: |x-y| + |y-z| = 2
so that row 1 of the array is (2,2,1), representing two 2s, two 3s, and one 4.
MATHEMATICA
t1[n_] := t1[n] = Tuples[Range[n], 3];
t[n_] := t[n] = Select[t1[n], #[[1]] < #[[2]] > #[[3]] &];
a[n_, k_] := Select[t[n], Abs[#[[1]] - #[[2]]] + Abs[#[[2]] - #[[3]]] == k &];
u = Table[Length[a[n, k]], {n, 1, 15}, {k, 2, n + 1}];
v = Flatten[u]
Column[Table[Length[a[n, k]], {n, 1, 15}, {k, 2, n + 1}]]
CROSSREFS
Cf. A000027 (column 1), A002620 (T(n,n)), A002717 (row sums), A368434, A368437, A368515, A368516, A368517, A368518, A368519, A368521, A368522.
Sequence in context: A368606 A331953 A113594 * A246425 A286468 A102563
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Jan 22 2024
STATUS
approved