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A368521
Triangular array T, read by rows: T(n,k) = number of sums |x-y| + |y-z| - |x-z| = k, where x,y,z are in {1,2,...,n}.
12
1, 6, 2, 17, 8, 2, 36, 18, 8, 2, 65, 32, 18, 8, 2, 106, 50, 32, 18, 8, 2, 161, 72, 50, 32, 18, 8, 2, 232, 98, 72, 50, 32, 18, 8, 2, 321, 128, 98, 72, 50, 32, 18, 8, 2, 430, 162, 128, 98, 72, 50, 32, 18, 8, 2, 561, 200, 162, 128, 98, 72, 50, 32, 18, 8, 2, 716
OFFSET
1,2
EXAMPLE
First eight rows:
1
6 2
17 8 2
36 18 8 2
65 32 18 8 2
106 50 32 18 8 2
161 72 50 32 18 8 2
232 98 72 50 32 18 8 2
For n=2, there are 8 triples (x,y,z):
111: |x-y| + |y-z| - |x-z| = 0
112: |x-y| + |y-z| - |x-z| = 0
121: |x-y| + |y-z| - |x-z| = 2
122: |x-y| + |y-z| - |x-z| = 0
211: |x-y| + |y-z| - |x-z| = 0
212: |x-y| + |y-z| - |x-z| = 2
221: |x-y| + |y-z| - |x-z| = 0
222: |x-y| + |y-z| - |x-z| = 0
so row 2 of the array is (6,2), representing six 0s and two 2s.
MATHEMATICA
t[n_] := t[n] = Tuples[Range[n], 3]
a[n_, k_] := Select[t[n], Abs[#[[1]] - #[[2]]] + Abs[#[[2]] - #[[3]]] - Abs[#[[1]] - #[[3]]] == k &]
u = Table[Length[a[n, k]], {n, 1, 15}, {k, 0, 2 n - 2, 2}]
v = Flatten[u] (* sequence *)
Column[Table[Length[a[n, k]], {n, 1, 15}, {k, 0, 2 n - 2, 2}]] (* array *)
CROSSREFS
Cf. A084990 (column 1), A000578 (row sums), A001105 (limiting reversed row), A368434, A368437, A368515, A368516, A368517, A368518, A368519, A368520, A368522, A368604, A368605, A368606, A368607, A368609.
Sequence in context: A136398 A376729 A228692 * A213771 A266711 A248269
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Jan 25 2024
STATUS
approved