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A368520
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 <= z.
10
1, 2, 2, 2, 3, 4, 7, 2, 2, 4, 6, 12, 8, 6, 2, 2, 5, 8, 17, 14, 15, 6, 6, 2, 2, 6, 10, 22, 20, 24, 16, 12, 6, 6, 2, 2, 7, 12, 27, 26, 33, 26, 25, 12, 12, 6, 6, 2, 2, 8, 14, 32, 32, 42, 36, 38, 26, 20, 12, 12, 6, 6, 2, 2, 9, 16, 37, 38, 51, 46, 51, 40, 37, 20, 20
OFFSET
1,2
COMMENTS
Row n consists of 2n-1 positive integers.
EXAMPLE
First seven rows:
1
2 2 2
3 4 7 2 2
4 6 12 8 6 2 2
5 8 17 14 15 6 6 2 2
6 10 22 20 24 16 12 6 6 2 2
7 12 27 26 33 26 25 12 12 6 6 2 2
For n=2, there are 6 triples (x,y,z) having x <= z:
111: |x-y| + |y-z| = 0
112: |x-y| + |y-z| = 1
121: |x-y| + |y-z| = 2
122: |x-y| + |y-z| = 1
212: |x-y| + |y-z| = 2
222: |x-y| + |y-z| = 0
so that row 1 of the array is (2,2,2), representing two 0s, two 1s, and two 2s.
MATHEMATICA
t1[n_] := t1[n] = Tuples[Range[n], 3];
t[n_] := t[n] = Select[t1[n], #[[1]] <= #[[3]] &];
a[n_, k_] := Select[t[n], Abs[#[[1]] - #[[2]]] + Abs[#[[2]] - #[[3]]] == k &];
u = Table[Length[a[n, k]], {n, 1, 15}, {k, 0, 2 n - 2}];
v = Flatten[u] (* sequence *)
Column[Table[Length[a[n, k]], {n, 1, 15}, {k, 0, 2 n - 2}]] (* array *)
CROSSREFS
Cf. A002411 (row sums), A110660 (limiting reverse row), A368434, A368437, A368515, A368516, A368517, A368518, A368519, A368521, A368522.
Sequence in context: A308620 A339711 A048185 * A377083 A095094 A275009
KEYWORD
nonn,tabf
AUTHOR
Clark Kimberling, Jan 22 2024
STATUS
approved