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A368516
Irregular 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.
11
2, 6, 4, 2, 10, 12, 8, 4, 2, 14, 20, 20, 12, 8, 4, 2, 18, 28, 32, 28, 18, 12, 8, 4, 2, 22, 36, 44, 44, 38, 24, 18, 12, 8, 4, 2, 26, 44, 56, 60, 58, 48, 32, 24, 18, 12, 8, 4, 2, 30, 52, 68, 76, 78, 72, 60, 40, 32, 24, 18, 12, 8, 4, 2, 34, 60, 80, 92, 98, 96
OFFSET
1,1
COMMENTS
Row n consists of 2n-1 even positive integers.
EXAMPLE
First six rows:
2
6 4 2
10 12 8 4 2
14 20 20 12 8 4 2
18 28 32 28 18 12 8 4 2
22 36 44 44 38 24 18 12 8 4 2
For n=3, there are 12 triples (x,y,z) having x != y and y != z:
121: |x-y| + |y-z| = 2
123: |x-y| + |y-z| = 2
131: |x-y| + |y-z| = 4
132: |x-y| + |y-z| = 3
212: |x-y| + |y-z| = 2
213: |x-y| + |y-z| = 3
231: |x-y| + |y-z| = 3
232: |x-y| + |y-z| = 2
312: |x-y| + |y-z| = 3
313: |x-y| + |y-z| = 4
321: |x-y| + |y-z| = 2
323: |x-y| + |y-z| = 2,
so that row 2 of the array is (6,4,2), representing six 2s, four 3s, and two 4s.
MATHEMATICA
t1[n_] := t1[n] = Tuples[Range[n], 3];
t[n_] := t[n] = Select[t1[n], #[[1]] != #[[2]] && #[[2]] != #[[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 - 2}]
v = Flatten[u]; (* sequence *)
Column[Table[Length[a[n, k]], {n, 2, 15}, {k, 2, 2 n - 2}]] (* array *)
CROSSREFS
Cf. A011379 (row sums), A007590 (limiting reverse row), A368434, A368437, A368515, A368517, A368518, A368519, A368520, A368521, A368522.
Sequence in context: A074382 A061350 A046276 * A283614 A333520 A003571
KEYWORD
nonn,tabf
AUTHOR
Clark Kimberling, Dec 31 2023
STATUS
approved