A353451 T(n,m) is the number of non-congruent quadrilaterals with integer vertex coordinates (x1,1), (n,y2), (x3,m), (1,y4), 1 < x1, x3 < n, 1 < y2, y4 < m, m <= n, such that the 6 distances between the 4 vertices are distinct and (x3-x1)*(y4-y2) = 0, where T(n,m) is a triangle read by rows.

0, 0, 0, 0, 2, 0, 1, 6, 6, 4, 1, 8, 10, 19, 12, 4, 15, 20, 39, 48, 40, 4, 16, 25, 41, 52, 89, 47, 10, 30, 39, 66, 86, 135, 144, 105, 10, 31, 41, 75, 91, 140, 142, 212, 106, 18, 49, 67, 107, 134, 203, 220, 308, 319, 214, 18, 49, 67, 109, 144, 210, 227, 325, 334, 458, 228

Offset: 3

Author

Rainer Rosenthal, May 13 2022

Keywords

• nonn
• tabl

Comments

Property "(x3-x1)*(y4-y2) = 0" holds iff one of the diagonals (spokes) of the quadrilateral is parallel to the x-axis or to the y-axis, i.e. not tilted (see example). The framed quadrilateral may be classified as "static" iff (x3-x1)*(y4-y2) = 0.

All quadrilaterals of A353532 are classified according to the sign of the product (x3-x1)*(y4-y2) as "all" = "unisense" (> 0) + "contrasense" (< 0) + "static" (= 0). The distinction is invariant under symmetry.

Examples

			The triangle begins
.
    \ m 3   4    5    6    7    8    9   10
   n \-------------------------------------
   3 |  0   |    |    |    |    |    |    |
   4 |  0,  0    |    |    |    |    |    |
   5 |  0,  2,   0    |    |    |    |    |
   6 |  1,  6,   6,   4    |    |    |    |
   7 |  1,  8,  10,  19,  12    |    |    |
   8 |  4, 15,  20,  39,  48,  40    |    |
   9 |  4, 16,  25,  41,  52,  89,  47    |
  10 | 10, 30,  39,  66,  86, 135, 144, 105
.
T(5,4) = a(5) = 2: See first 2 examples for (5,4) in A353532.
  .
     4 | . C . . .
     3 | . . . . .     A = (x1,1) = (2,1), B = (5,y2) = (5,2)
     2 | D . . . B     C = (x3,4) = (2,4), D = (1,y4) = (1,2)
     1 | . A . . .
     y /----------      (x3-x1) * (y4-y2) = (2-2)*(2-2) = 0
       x 1 2 3 4 5
  .
     4 | . C . . .
     3 | . . . . B     A = (x1,1) = (2,1), B = (5,y2) = (5,3)
     2 | D . . . .     C = (x3,4) = (2,4), D = (1,y4) = (1,2)
     1 | . A . . .
     y /----------      (x3-x1) * (y4-y2) = (2-2)*(2-3) = 0
       x 1 2 3 4 5
  .
T(5,4) = 2 since these are the only static configurations of A353532(5,4). Spoke AC is not tilted, but parallel to the y-axis. First example: spoke DB is not tilted, but parallel to the x-axis. Second example: spoke DB is not parallel to the x-axis, but tilted to the left. We have (x3-x1)*(y4-y2) = 0 in both cases, so these framed quadrilaterals have the "static" property.
		

Links

• Rainer Rosenthal, Rows n = 3..100, flattened

Crossrefs

Cf. A353532 ("all"), A353449 ("unisense"), A353450 ("contrasense").