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A368515
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.
11
2, 2, 4, 8, 4, 2, 6, 14, 14, 8, 4, 2, 8, 20, 24, 22, 12, 8, 4, 2, 10, 26, 34, 36, 30, 18, 12, 8, 4, 2, 12, 32, 44, 50, 48, 40, 24, 18, 12, 8, 4, 2, 14, 38, 54, 64, 66, 62, 50, 32, 24, 18, 12, 8, 4, 2, 16, 44, 64, 78, 84, 84, 76, 62, 40, 32, 24, 18, 12, 8, 4
OFFSET
1,1
COMMENTS
Row n consists of 2n even positive integers.
EXAMPLE
First six rows:
2 2
4 8 4 2
6 14 14 8 4 2
8 20 24 22 12 8 4 2
10 26 34 36 30 18 12 8 4 2
12 32 44 50 48 40 24 18 12 8 4 2
For n=2, there are 4 triples (x,y,z) having x != y:
121: |x-y| + |y-z| = 2
122: |x-y| + |y-z| = 1
211: |x-y| + |y-z| = 1
212: |x-y| + |y-z| = 2,
so that row 2 of the array is (2,2), representing two 1s and two 2s.
MATHEMATICA
t1[n_] := t1[n] = Tuples[Range[n], 3];
t[n_] := t[n] = Select[t1[n], #[[1]] != #[[2]] &];
a[n_, k_] := Select[t[n], Abs[#[[1]] - #[[2]]] + Abs[#[[2]] - #[[3]]] == k &];
u = Table[Length[a[n, k]], {n, 2, 15}, {k, 1, 2 n - 2}];
v = Flatten[u]; (* sequence *)
Column[Table[Length[a[n, k]], {n, 2, 15}, {k, 1, 2 n - 2}]] (* array *)
CROSSREFS
Cf. A045991 (row sums), A007590 (limiting reverse row), A368434, A368437, A368516, A368517, A368518, A368519, A368520, A368521, A368522.
Sequence in context: A045777 A136534 A121175 * A366714 A359488 A183397
KEYWORD
nonn,tabf
AUTHOR
Clark Kimberling, Dec 31 2023
STATUS
approved