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.

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.

Links

Table of n, a(n) for n=1..78.

Hugo Pfoertner, PARI program

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

Adjacent sequences: A354696 A354697 A354698 * A354700 A354701 A354702

Keyword

nonn,tabl

Author

Hugo Pfoertner, Jun 07 2022

Status

approved