proposed

approved

editing

proposed

allocated for Clark KimberlingIrregular 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.

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

1,2

Row n consists of 2n-1 positive integers.

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.

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 *)

Cf. A002411 (row sums), A110660 (limiting reverse row), A368434, A368437, A368515, A368516, A368517, A368518, A368519, A368521, A368522.

allocated

nonn,tabf

Clark Kimberling, Jan 22 2024

approved

editing

allocated for Clark Kimberling

allocated

approved