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A368608
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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.
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0
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2, 1, 4, 5, 2, 1, 6, 9, 8, 4, 2, 1, 8, 13, 14, 12, 6, 4, 2, 1, 10, 17, 20, 20, 16, 9, 6, 4, 2, 1, 12, 21, 26, 28, 26, 21, 12, 9, 6, 4, 2, 1, 14, 25, 32, 36, 36, 33, 26, 16, 12, 9, 6, 4, 2, 1, 16, 29, 38, 44, 46, 45, 40, 32, 20, 16, 12, 9, 6, 4, 2, 1, 18, 33
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OFFSET
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1,1
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COMMENTS
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Row n consists of 2n positive integers.
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LINKS
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EXAMPLE
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First six rows:
2 1
4 5 2 1
6 9 8 4 2 1
8 13 14 12 6 4 2 1
10 17 20 20 16 9 6 4 2 1
12 21 26 28 26 21 12 9 6 4 2 1
For n=2, there are 3 triples (x,y,z) having x != y and y <= z:
122: |x-y| + |y-z| = 1
211: |x-y| + |y-z| = 1
212: |x-y| + |y-z| = 2
so row 2 of the array is (2,1), representing two 1s and one 2.
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MATHEMATICA
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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, 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 *)
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CROSSREFS
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Cf. A005443 (column 1), A027480 (row sums), A002620 (limiting reversed row), A368434, A368437, A368515, A368516, A368517, A368518, A368519, A368520, A368521, A368522, A368604, A368605, A368606, A368607, A368609.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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