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%I #7 Jun 08 2022 15:38:42
%S 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,
%T 12,13,13,15,16,17,10,13,14,14,16,17,18,19,20,12,14,15,16,17,18,18,20,
%U 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
%N 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.
%C T(w,h) is a triangle, read by rows.
%H Hugo Pfoertner, <a href="/A354699/a354699.gp.txt">PARI program</a>
%e The triangle begins:
%e 0;
%e 2, 1;
%e 4, 5, 3;
%e 5, 6, 7, 4;
%e 7, 8, 8, 9, 6;
%e 8, 9, 10, 11, 12, 7;
%e 10, 11, 12, 13, 14, 15, 9;
%e 11, 12, 13, 13, 15, 16, 17, 10;
%e 13, 14, 14, 16, 17, 18, 19, 20, 12;
%e 14, 15, 16, 17, 18, 18, 20, 20, 22, 13
%e .
%e T(2,1) = 2:
%e 1 | . . C Squared sides s^2:
%e 0 | A B . AB = 1, BC = 2, CA = 5
%e y /------
%e x 0 1 2
%e 1 | . . C
%e 0 | A . B AB = 4, BC = 1, CA = 5
%e y /------
%e x 0 1 2
%e .
%e T(2,2) = 1:
%e 2 | . . C
%e 1 | . . . Squared sides s^2:
%e 0 | A B . AB = 1, BC = 5, CA = 8
%e y /------
%e x 0 1 2
%e .
%e T(3,1) = 4:
%e 1 | . . . C 1 | . . . C 1 | . . . C 1 | . C . .
%e 0 | A B . . 0 | A . B . 0 | A . . B 0 | A . . B
%e y /-------- y /-------- y /-------- y /--------
%e x 0 1 2 3 x 0 1 2 3 x 0 1 2 3 x 0 1 2 3
%e s^2: {1,5,10} {2,4,10} {1,9,10} {2,5,9}
%e .
%e T(3,2) = 5:
%e 2 | . . . C 2 | . . . C 2 | . . . C 2 | . . . C 2 | . C . .
%e 1 | . . . . 1 | . . . . 1 | . . . . 1 | . . . B 1 | . . . .
%e 0 | A B . . 0 | A . B . 0 | A . . B 0 | A . . . 0 | A . . B
%e y /-------- y /-------- y /-------- y /-------- y /--------
%e 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
%e s^2: {1,8,13} {4,5,13} {4,9,13} {1,10,13} {5,8,9}
%e .
%e T(3,3) = 3:
%e 3 | . . . C 3 | . . . C 3 | . C . .
%e 2 | . . . . 2 | . . . . 2 | . . . .
%e 1 | . . . . 1 | . . . . 1 | . . . .
%e 0 | A B . . 0 | A . B . 0 | A . . B
%e y /-------- y /-------- y /--------
%e x 0 1 2 3 x 0 1 2 3 x 0 1 2 3
%e s^2: {1,13,18} {4,10,18} {9,10,13}
%o (PARI) see link.
%Y Cf. A354700.
%K nonn,tabl
%O 1,2
%A _Hugo Pfoertner_, Jun 07 2022
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