A368522 -id:A368522 - cp's OEIS frontend

A368158 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.

2, 3, 1, 3, 6, 6, 2, 1, 4, 9, 11, 9, 4, 2, 1, 5, 12, 16, 16, 13, 6, 4, 2, 1, 6, 15, 21, 23, 22, 17, 9, 6, 4, 2, 1, 7, 18, 26, 30, 31, 28, 22, 12, 9, 6, 4, 2, 1, 8, 21, 31, 37, 40, 39, 35, 27, 16, 12, 9, 6, 4, 2, 1, 9, 24, 36, 44, 49, 50, 48, 42, 33, 20, 16

Offset: 1

Clark Kimberling, Jan 20 2024

Row n consists of 2n+1 positive integers.

			First six rows:
  2   3   1
  3   6   6   2   1
  4   9  11   9   4   2   1
  5  12  16  16  13   6   4   2  1
  6  15  21  23  22  17   9   6  4  2  1
  7  18  26  30  31  28  22  12  9  6  4  2  1
For n=2, there are 6 triples (x,y,z) having x <= y:
  111:  |x-y| + |y-z| = 0
  112:  |x-y| + |y-z| = 1
  121:  |x-y| + |y-z| = 2
  122:  |x-y| + |y-z| = 1
  221:  |x-y| + |y-z| = 1
  222:  |x-y| + |y-z| = 0,
so row 1 of the array is (2,3,1), representing two 0s, three 1s, and one 1.

Cf. A002411 (row sums), A002620 (limiting reverse row), A368434, A368437, A368515, A368516, A368517, A368519, A368520, A368521, A368522.

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, 0, 2 n - 2}];
v = Flatten[u]  (* sequence *)
Column[Table[Length[a[n, k]], {n, 2, 15}, {k, 0, 2 n - 2}]]  (* array *)

A368608 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.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jan 25 2024

Row n consists of 2n positive integers.

			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.

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.

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 *)

cp's OEIS Frontend

A368158 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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica