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.

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.

Links

Table of n, a(n) for n=1..75.

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

Adjacent sequences: A368517 A368518 A368519 * A368521 A368522 A368523

Keyword

nonn,tabf

Author

Clark Kimberling, Jan 22 2024

Status

approved