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A354699
T(w,h) is the number of non-congruent triangles with distinct side lengths whose vertices with integer coordinates (x_i, y_i) all lie on the perimeter of a rectangle of width w and height h, with max(x_i)-min(x_i) = w and max(y_i)-min(y_i) = h.
2
0, 2, 1, 4, 5, 3, 5, 6, 7, 4, 7, 8, 8, 9, 6, 8, 9, 10, 11, 12, 7, 10, 11, 12, 13, 14, 15, 9, 11, 12, 13, 13, 15, 16, 17, 10, 13, 14, 14, 16, 17, 18, 19, 20, 12, 14, 15, 16, 17, 18, 18, 20, 20, 22, 13, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 16
OFFSET
1,2
COMMENTS
T(w,h) is a triangle, read by rows.
EXAMPLE
The triangle begins:
0;
2, 1;
4, 5, 3;
5, 6, 7, 4;
7, 8, 8, 9, 6;
8, 9, 10, 11, 12, 7;
10, 11, 12, 13, 14, 15, 9;
11, 12, 13, 13, 15, 16, 17, 10;
13, 14, 14, 16, 17, 18, 19, 20, 12;
14, 15, 16, 17, 18, 18, 20, 20, 22, 13
.
T(2,1) = 2:
1 | . . C Squared sides s^2:
0 | A B . AB = 1, BC = 2, CA = 5
y /------
x 0 1 2
1 | . . C
0 | A . B AB = 4, BC = 1, CA = 5
y /------
x 0 1 2
.
T(2,2) = 1:
2 | . . C
1 | . . . Squared sides s^2:
0 | A B . AB = 1, BC = 5, CA = 8
y /------
x 0 1 2
.
T(3,1) = 4:
1 | . . . C 1 | . . . C 1 | . . . C 1 | . C . .
0 | A B . . 0 | A . B . 0 | A . . B 0 | A . . B
y /-------- y /-------- y /-------- y /--------
x 0 1 2 3 x 0 1 2 3 x 0 1 2 3 x 0 1 2 3
s^2: {1,5,10} {2,4,10} {1,9,10} {2,5,9}
.
T(3,2) = 5:
2 | . . . C 2 | . . . C 2 | . . . C 2 | . . . C 2 | . C . .
1 | . . . . 1 | . . . . 1 | . . . . 1 | . . . B 1 | . . . .
0 | A B . . 0 | A . B . 0 | A . . B 0 | A . . . 0 | A . . B
y /-------- y /-------- y /-------- y /-------- y /--------
x 0 1 2 3 x 0 1 2 3 x 0 1 2 3 x 0 1 2 3 x 0 1 2 3
s^2: {1,8,13} {4,5,13} {4,9,13} {1,10,13} {5,8,9}
.
T(3,3) = 3:
3 | . . . C 3 | . . . C 3 | . C . .
2 | . . . . 2 | . . . . 2 | . . . .
1 | . . . . 1 | . . . . 1 | . . . .
0 | A B . . 0 | A . B . 0 | A . . B
y /-------- y /-------- y /--------
x 0 1 2 3 x 0 1 2 3 x 0 1 2 3
s^2: {1,13,18} {4,10,18} {9,10,13}
PROG
(PARI) see link.
CROSSREFS
Cf. A354700.
Sequence in context: A099492 A359708 A144203 * A239806 A330402 A075302
KEYWORD
nonn,tabl
AUTHOR
Hugo Pfoertner, Jun 07 2022
STATUS
approved