A368158 - OEIS

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

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OFFSET

1,1

COMMENTS

Row n consists of 2n+1 positive integers.

LINKS

Table of n, a(n) for n=1..74.

EXAMPLE

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.

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, 0, 2 n - 2}];

v = Flatten[u] (* sequence *)

Column[Table[Length[a[n, k]], {n, 2, 15}, {k, 0, 2 n - 2}]] (* array *)

CROSSREFS

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

Sequence in context: A101912 A208522 A209569 * A176850 A208516 A111808

Adjacent sequences: A368155 A368156 A368157 * A368159 A368160 A368161

KEYWORD

nonn,tabf

AUTHOR

Clark Kimberling, Jan 20 2024

STATUS

approved