A368520 -id:A368520

A368521

Triangular array T, read by rows: T(n,k) = number of sums |x-y| + |y-z| - |x-z| = k, where x,y,z are in {1,2,...,n}.

+0
12

1, 6, 2, 17, 8, 2, 36, 18, 8, 2, 65, 32, 18, 8, 2, 106, 50, 32, 18, 8, 2, 161, 72, 50, 32, 18, 8, 2, 232, 98, 72, 50, 32, 18, 8, 2, 321, 128, 98, 72, 50, 32, 18, 8, 2, 430, 162, 128, 98, 72, 50, 32, 18, 8, 2, 561, 200, 162, 128, 98, 72, 50, 32, 18, 8, 2, 716

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..67.

EXAMPLE

First eight rows:

1

6 2

17 8 2

36 18 8 2

65 32 18 8 2

106 50 32 18 8 2

161 72 50 32 18 8 2

232 98 72 50 32 18 8 2

For n=2, there are 8 triples (x,y,z):

111: |x-y| + |y-z| - |x-z| = 0

112: |x-y| + |y-z| - |x-z| = 0

121: |x-y| + |y-z| - |x-z| = 2

122: |x-y| + |y-z| - |x-z| = 0

211: |x-y| + |y-z| - |x-z| = 0

212: |x-y| + |y-z| - |x-z| = 2

221: |x-y| + |y-z| - |x-z| = 0

222: |x-y| + |y-z| - |x-z| = 0

so row 2 of the array is (6,2), representing six 0s and two 2s.

MATHEMATICA

t[n_] := t[n] = Tuples[Range[n], 3]

a[n_, k_] := Select[t[n], Abs[#[[1]] - #[[2]]] + Abs[#[[2]] - #[[3]]] - Abs[#[[1]] - #[[3]]] == k &]

u = Table[Length[a[n, k]], {n, 1, 15}, {k, 0, 2 n - 2, 2}]

v = Flatten[u] (* sequence *)

Column[Table[Length[a[n, k]], {n, 1, 15}, {k, 0, 2 n - 2, 2}]] (* array *)

CROSSREFS

Cf. A084990 (column 1), A000578 (row sums), A001105 (limiting reversed row), A368434, A368437, A368515, A368516, A368517, A368518, A368519, A368520, A368522, A368604, A368605, A368606, A368607, A368609.

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Jan 25 2024

STATUS

approved

A368515

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.

+0
11

2, 2, 4, 8, 4, 2, 6, 14, 14, 8, 4, 2, 8, 20, 24, 22, 12, 8, 4, 2, 10, 26, 34, 36, 30, 18, 12, 8, 4, 2, 12, 32, 44, 50, 48, 40, 24, 18, 12, 8, 4, 2, 14, 38, 54, 64, 66, 62, 50, 32, 24, 18, 12, 8, 4, 2, 16, 44, 64, 78, 84, 84, 76, 62, 40, 32, 24, 18, 12, 8, 4

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Row n consists of 2n even positive integers.

LINKS

Table of n, a(n) for n=1..71.

EXAMPLE

First six rows:

2 2

4 8 4 2

6 14 14 8 4 2

8 20 24 22 12 8 4 2

10 26 34 36 30 18 12 8 4 2

12 32 44 50 48 40 24 18 12 8 4 2

For n=2, there are 4 triples (x,y,z) having x != y:

121: |x-y| + |y-z| = 2

122: |x-y| + |y-z| = 1

211: |x-y| + |y-z| = 1

212: |x-y| + |y-z| = 2,

so that row 2 of the array is (2,2), representing two 1s and two 2s.

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, 1, 2 n - 2}];

v = Flatten[u]; (* sequence *)

Column[Table[Length[a[n, k]], {n, 2, 15}, {k, 1, 2 n - 2}]] (* array *)

CROSSREFS

Cf. A045991 (row sums), A007590 (limiting reverse row), A368434, A368437, A368516, A368517, A368518, A368519, A368520, A368521, A368522.

KEYWORD

nonn,tabf

AUTHOR

Clark Kimberling, Dec 31 2023

STATUS

approved

A368516

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.

+0
11

2, 6, 4, 2, 10, 12, 8, 4, 2, 14, 20, 20, 12, 8, 4, 2, 18, 28, 32, 28, 18, 12, 8, 4, 2, 22, 36, 44, 44, 38, 24, 18, 12, 8, 4, 2, 26, 44, 56, 60, 58, 48, 32, 24, 18, 12, 8, 4, 2, 30, 52, 68, 76, 78, 72, 60, 40, 32, 24, 18, 12, 8, 4, 2, 34, 60, 80, 92, 98, 96

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Row n consists of 2n-1 even positive integers.

LINKS

Table of n, a(n) for n=1..70.

EXAMPLE

First six rows:

2

6 4 2

10 12 8 4 2

14 20 20 12 8 4 2

18 28 32 28 18 12 8 4 2

22 36 44 44 38 24 18 12 8 4 2

For n=3, there are 12 triples (x,y,z) having x != y and y != z:

121: |x-y| + |y-z| = 2

123: |x-y| + |y-z| = 2

131: |x-y| + |y-z| = 4

132: |x-y| + |y-z| = 3

212: |x-y| + |y-z| = 2

213: |x-y| + |y-z| = 3

231: |x-y| + |y-z| = 3

232: |x-y| + |y-z| = 2

312: |x-y| + |y-z| = 3

313: |x-y| + |y-z| = 4

321: |x-y| + |y-z| = 2

323: |x-y| + |y-z| = 2,

so that row 2 of the array is (6,4,2), representing six 2s, four 3s, and two 4s.

MATHEMATICA

t1[n_] := t1[n] = Tuples[Range[n], 3];

t[n_] := t[n] = Select[t1[n], #[[1]] != #[[2]] && #[[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, 2, 2 n - 2}]

v = Flatten[u]; (* sequence *)

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

CROSSREFS

Cf. A011379 (row sums), A007590 (limiting reverse row), A368434, A368437, A368515, A368517, A368518, A368519, A368520, A368521, A368522.

KEYWORD

nonn,tabf

AUTHOR

Clark Kimberling, Dec 31 2023

STATUS

approved

A368517

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.

+0
11

1, 1, 2, 4, 2, 1, 3, 7, 7, 4, 2, 1, 4, 10, 12, 11, 6, 4, 2, 1, 5, 13, 17, 18, 15, 9, 6, 4, 2, 1, 6, 16, 22, 25, 24, 20, 12, 9, 6, 4, 2, 1, 7, 19, 27, 32, 33, 31, 25, 16, 12, 9, 6, 4, 2, 1, 8, 22, 32, 39, 42, 42, 38, 31, 20, 16, 12, 9, 6, 4, 2, 1, 9, 25, 37

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

Row n consists of 2n positive integers.

LINKS

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

EXAMPLE

First eight rows:

1 1

2 4 2 1

3 7 7 4 2 1

4 10 12 11 6 4 2 1

5 13 17 18 15 9 6 4 2 1

6 16 22 25 24 20 12 9 6 4 2 1

7 19 27 32 33 31 25 16 12 9 6 4 2 1

8 22 32 39 42 42 38 31 20 16 12 9 6 4 2 1

For n=3, there are 9 triples (x,y,z) having x < y:

121: |x-y| + |y-z| = 2

122: |x-y| + |y-z| = 1

123: |x-y| + |y-z| = 2

131: |x-y| + |y-z| = 4

132: |x-y| + |y-z| = 3

133: |x-y| + |y-z| = 2

231: |x-y| + |y-z| = 3

232: |x-y| + |y-z| = 2

233: |x-y| + |y-z| = 1,

so that row 2 of the array is (2,4,2,1), representing two 1s, four 2s, two 3s, and one 4.

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, 1, 2 n - 2}];

v = Flatten[u] (* sequence *)

Column[Table[Length[a[n, k]], {n, 2, 15}, {k, 1, 2 n - 2}]] (* array *)

CROSSREFS

Cf. A006002 (row sums), A002620 (limiting reverse row), A368434, A368437, A368515, A368516, A368518, A368519, A368520, A368521, A368522.

KEYWORD

nonn,tabf

AUTHOR

Clark Kimberling, Dec 31 2023

STATUS

approved

A368519

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.

+0
11

2, 4, 3, 2, 6, 6, 8, 2, 2, 8, 9, 14, 9, 6, 2, 2, 10, 12, 20, 16, 16, 6, 6, 2, 2, 12, 15, 26, 23, 26, 17, 12, 6, 6, 2, 2, 14, 18, 32, 30, 36, 28, 26, 12, 12, 6, 6, 2, 2, 16, 21, 38, 37, 46, 39, 40, 27, 20, 12, 12, 6, 6, 2, 2, 18, 24, 44, 44, 56, 50, 54, 42

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Row n consists of 2n-1 positive integers.

LINKS

Table of n, a(n) for n=1..72.

EXAMPLE

First six rows:

2

4 3 2

6 6 8 2 2

8 9 14 9 6 2 2

10 12 20 16 16 6 6 2 2

12 15 26 23 26 17 12 6 6 2 2

For n=3, there are 9 triples (x,y,z) having x < z:

112: |x-y| + |y-z| = 1

113: |x-y| + |y-z| = 2

122: |x-y| + |y-z| = 1

123: |x-y| + |y-z| = 2

132: |x-y| + |y-z| = 3

133: |x-y| + |y-z| = 2

213: |x-y| + |y-z| = 3

223: |x-y| + |y-z| = 1

233: |x-y| + |y-z| = 1,

so that row 1 of the array is (4,3,2), representing four 1s, three 2s, and two 3s.

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, 2, 15}, {k, 1, 2 n - 3}];

v = Flatten[u]; (* sequence *)

Column[Table[Length[a[n, k]], {n, 2, 15}, {k, 1, 2 n - 3}]] (* array *)

CROSSREFS

Cf. A006002 (row sums), A110660 (limiting reverse row), A368434, A368437, A368515, A368516, A368517, A368518, A368520, A368521, A368522.

KEYWORD

nonn,tabf

AUTHOR

Clark Kimberling, Jan 22 2024

STATUS

approved

A368605

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.

+0
4

1, 1, 2, 3, 2, 1, 3, 5, 5, 4, 2, 1, 4, 7, 8, 8, 6, 4, 2, 1, 5, 9, 11, 12, 11, 9, 6, 4, 2, 1, 6, 11, 14, 16, 16, 15, 12, 9, 6, 4, 2, 1, 7, 13, 17, 20, 21, 21, 19, 16, 12, 9, 6, 4, 2, 1, 8, 15, 20, 24, 26, 27, 26, 24, 20, 16, 12, 9, 6, 4, 2, 1, 9, 17, 23, 28

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

Row n consists of 2n positive integers.

LINKS

Table of n, a(n) for n=1..76.

EXAMPLE

First six rows:

1 1

2 3 2 1

3 5 5 4 2 1

4 7 8 8 6 4 2 1

5 9 11 12 11 9 6 4 2 1

6 11 14 16 16 15 12 9 6 4 2 1

For n=3, there are 8 triples (x,y,z) having x < y and y >= z:

121: |x-y| + |y-z| = 2

122: |x-y| + |y-z| = 1

131: |x-y| + |y-z| = 4

132: |x-y| + |y-z| = 3

133: |x-y| + |y-z| = 2

231: |x-y| + |y-z| = 3

232: |x-y| + |y-z| = 2

233: |x-y| + |y-z| = 1

so row 1 of the array is (2,3,2,1), representing two 1s, three 2s, two 3s, and one 4.

MATHEMATICA

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 *)

CROSSREFS

Cf. A000027 (column 1), A007290 (row sums), A002620 (limiting reversed row), A368434, A368437, A368515, A368516, A368517, A368518, A368519, A368520, A368521, A368522, A368604, A368606, A368607, A368609.

KEYWORD

nonn,tabf

AUTHOR

Clark Kimberling, Jan 22 2024

STATUS

approved

A368606

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.

+0
4

1, 2, 2, 1, 3, 4, 4, 2, 1, 4, 6, 7, 6, 4, 2, 1, 5, 8, 10, 10, 9, 6, 4, 2, 1, 6, 10, 13, 14, 14, 12, 9, 6, 4, 2, 1, 7, 12, 16, 18, 19, 18, 16, 12, 9, 6, 4, 2, 1, 8, 14, 19, 22, 24, 24, 23, 20, 16, 12, 9, 6, 4, 2, 1, 9, 16, 22, 26, 29, 30, 30, 28, 25, 20, 16

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Row n consists of 2n-1 positive integers.

LINKS

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

EXAMPLE

First six rows:

1

2 2 1

3 4 4 2 1

4 6 7 6 4 2 1

5 8 10 10 9 6 4 2 1

6 10 13 14 14 12 9 6 4 2 1

For n=2, there are 5 triples (x,y,z) having x <= y and y >= z:

111: |x-y| + |y-z| = 0

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 2 of the array is (2,2,1), representing two 0s, two 1s, and one 3.

MATHEMATICA

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, 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. A000027 (column 1), A000330 (row sums), A002620 (limiting reversed row), A368434, A368437, A368515, A368516, A368517, A368518, A368519, A368520, A368521, A368522, A368604, A368605, A368607, A368609.

KEYWORD

nonn,tabf

AUTHOR

Clark Kimberling, Jan 22 2024

STATUS

approved

A368607

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.

+0
4

1, 3, 2, 1, 5, 6, 4, 2, 1, 7, 10, 10, 6, 4, 2, 1, 9, 14, 16, 14, 9, 6, 4, 2, 1, 11, 18, 22, 22, 19, 12, 9, 6, 4, 2, 1, 13, 22, 28, 30, 29, 24, 16, 12, 9, 6, 4, 2, 1, 15, 26, 34, 38, 39, 36, 30, 20, 16, 12, 9, 6, 4, 2, 1, 17, 30, 40, 46, 49, 48, 44, 36, 25

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Row n consists of 2n-1 positive integers.

LINKS

Table of n, a(n) for n=1..73.

EXAMPLE

First six rows:

1

3 2 1

5 6 4 2 1

7 10 10 6 4 2 1

9 14 16 14 9 6 4 2 1

11 18 22 22 19 12 9 6 4 2 1

For n=3, there are 6 triples (x,y,z) having x != y and y < z:

123: |x-y| + |y-z| = 2

212: |x-y| + |y-z| = 2

213: |x-y| + |y-z| = 3

312: |x-y| + |y-z| = 3

313: |x-y| + |y-z| = 4

323: |x-y| + |y-z| = 2

so row 2 of the array is (3,2,1), representing three 2s, two 3s, and one 4.

MATHEMATICA

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, 2, 2 n - 2}];

v = Flatten[u] (* sequence *)

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

CROSSREFS

Cf. A005408 (column 1), A002411 (row sums), A002620 (limiting reversed row), A368434, A368437, A368515, A368516, A368517, A368518, A368519, A368520, A368521, A368522, A368604, A368605, A368606, A368609.

KEYWORD

nonn,tabf

AUTHOR

Clark Kimberling, Jan 25 2024

STATUS

approved

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.

+0
0

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

(list; graph; refs; listen; history; text; internal format)

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.

KEYWORD

nonn,tabf

AUTHOR

Clark Kimberling, Jan 20 2024

STATUS

approved

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.

+0
0

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Row n consists of 2n positive integers.

LINKS

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

EXAMPLE

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.

MATHEMATICA

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 *)

CROSSREFS

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.

KEYWORD

nonn,tabf

AUTHOR

Clark Kimberling, Jan 25 2024

STATUS

approved