cp's OEIS Frontend

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.

%I A354699 #8 Dec 19 2024 11:53:22
%S A354699 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 A354699 12,13,13,15,16,17,10,13,14,14,16,17,18,19,20,12,14,15,16,17,18,18,20,
%U A354699 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 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.
%C A354699 T(w,h) is a triangle, read by rows.
%H A354699 Hugo Pfoertner, <a href="/A354699/a354699.gp.txt">PARI program</a>
%e A354699 The triangle begins:
%e A354699    0;
%e A354699    2,  1;
%e A354699    4,  5,  3;
%e A354699    5,  6,  7,  4;
%e A354699    7,  8,  8,  9,  6;
%e A354699    8,  9, 10, 11, 12,  7;
%e A354699   10, 11, 12, 13, 14, 15, 9;
%e A354699   11, 12, 13, 13, 15, 16, 17, 10;
%e A354699   13, 14, 14, 16, 17, 18, 19, 20, 12;
%e A354699   14, 15, 16, 17, 18, 18, 20, 20, 22, 13
%e A354699 .
%e A354699 T(2,1) = 2:
%e A354699   1 | . . C  Squared sides s^2:
%e A354699   0 | A B .  AB = 1, BC = 2, CA = 5
%e A354699   y /------
%e A354699     x 0 1 2
%e A354699   1 | . . C
%e A354699   0 | A . B  AB = 4, BC = 1, CA = 5
%e A354699   y /------
%e A354699     x 0 1 2
%e A354699 .
%e A354699 T(2,2) = 1:
%e A354699   2 | . . C
%e A354699   1 | . . .  Squared sides s^2:
%e A354699   0 | A B .  AB = 1, BC = 5, CA = 8
%e A354699   y /------
%e A354699     x 0 1 2
%e A354699 .
%e A354699 T(3,1) = 4:
%e A354699     1 | . . . C    1 | . . . C    1 | . . . C    1 | . C . .
%e A354699     0 | A B . .    0 | A . B .    0 | A . . B    0 | A . . B
%e A354699     y /--------    y /--------    y /--------    y /--------
%e A354699       x 0 1 2 3      x 0 1 2 3      x 0 1 2 3      x 0 1 2 3
%e A354699   s^2: {1,5,10}       {2,4,10}       {1,9,10}       {2,5,9}
%e A354699 .
%e A354699 T(3,2) = 5:
%e A354699     2 | . . . C   2 | . . . C   2 | . . . C   2 | . . . C   2 | . C . .
%e A354699     1 | . . . .   1 | . . . .   1 | . . . .   1 | . . . B   1 | . . . .
%e A354699     0 | A B . .   0 | A . B .   0 | A . . B   0 | A . . .   0 | A . . B
%e A354699     y /--------   y /--------   y /--------   y /--------   y /--------
%e A354699       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 A354699   s^2: {1,8,13}      {4,5,13}      {4,9,13}     {1,10,13}      {5,8,9}
%e A354699 .
%e A354699 T(3,3) = 3:
%e A354699     3 | . . . C   3 | . . . C   3 | . C . .
%e A354699     2 | . . . .   2 | . . . .   2 | . . . .
%e A354699     1 | . . . .   1 | . . . .   1 | . . . .
%e A354699     0 | A B . .   0 | A . B .   0 | A . . B
%e A354699     y /--------   y /--------   y /--------
%e A354699       x 0 1 2 3     x 0 1 2 3     x 0 1 2 3
%e A354699   s^2: {1,13,18}    {4,10,18}     {9,10,13}
%o A354699 (PARI) \\ See link.
%Y A354699 Cf. A354700.
%K A354699 nonn,tabl
%O A354699 1,2
%A A354699 _Hugo Pfoertner_, Jun 07 2022