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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.
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%I #28 Jun 09 2022 11:40:50

%S 0,0,0,0,3,1,1,7,12,11,1,11,26,52,40,4,23,50,94,147,105,4,30,69,127,

%T 198,301,190,10,49,103,192,302,444,583,379,10,58,127,244,387,576,754,

%U 1039,616,18,84,180,329,509,756,989,1334,1680,987,18,94,209,389,611,910,1203,1618,2052,2581,1426

%N 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.

%C T(n,m) is a triangle, read by rows.

%H Rainer Rosenthal, <a href="/A353532/b353532.txt">Rows n = 3..100, flattened</a>

%H Hugo Pfoertner, <a href="/A353532/a353532.gp.txt">PARI program</a>

%e The triangle begins

%e \ m 3 4 5 6 7 8 9 10

%e n \-------------------------------------

%e 3 | 0, | | | | | | |

%e 4 | 0, 0, | | | | | |

%e 5 | 0, 3, 1, | | | | |

%e 6 | 1, 7, 12, 11, | | | |

%e 7 | 1, 11, 26, 52, 40, | | |

%e 8 | 4, 23, 50, 94, 147, 105, | |

%e 9 | 4, 30, 69, 127, 198, 301, 190, |

%e 10 | 10, 49, 103, 192, 302, 444, 583, 379

%e .

%e .

%e 4 | . C . . . There are six squared distances.

%e 3 | . . . . . They are arranged as follows:

%e 2 | D . . . B AB-BC-CD-DA (counterclockwise)

%e 1 | . A . . . AC X DB (across)

%e y /---------- Here: AB = 3^2 + 1^2 = 10,

%e x 1 2 3 4 5 BC = 13, CD = 5, DA = 2,

%e . AC = 9, DB = 16

%e 10-13-5-2 <==== yielding this

%e 9 X 16 <==== description

%e .

%e .

%e T(5,4) = a(5) = 3:

%e .

%e 4 | . X . . . 4 | . X . . . 4 | . . X . .

%e 3 | . . . . . 3 | . . . . X 3 | . . . . X

%e 2 | X . . . X 2 | X . . . . 2 | X . . . .

%e 1 | . X . . . 1 | . X . . . 1 | . X . . .

%e y /---------- y /---------- y /----------

%e x 1 2 3 4 5 x 1 2 3 4 5 x 1 2 3 4 5

%e .

%e 10-13-5-2 13-10-5-2 13-5-8-2

%e 9 X 16 9 X 17 10 X 17

%e .

%e T(5,5) = a(6) = A353447(5) = 1:

%e .

%e 5 | . . . X .

%e 4 | . . . . .

%e 3 | . . . . X 13-5-18-2

%e 2 | X . . . . 20 X 17

%e 1 | . X . . .

%e y /----------

%e x 1 2 3 4 5

%e .

%e T(6,3) = a(7) = 1:

%e .

%e 3 | . . . X . .

%e 2 | X . . . . X 17-5-10-2

%e 1 | . X . . . . 8 X 25

%e y /------------

%e x 1 2 3 4 5 6

%e .

%e T(6,4) = a(8) = 7:

%e .

%e 4 | . X . . . . 4 | . X . . . . 4 | . . X . . . 4 | . . . X . .

%e 3 | . . . . . . 3 | . . . . . X 3 | . . . . . . 3 | X . . . . .

%e 2 | X . . . . X 2 | X . . . . . 2 | X . . . . X 2 | . . . . . X

%e 1 | . X . . . . 1 | . X . . . . 1 | . X . . . . 1 | . X . . . .

%e y /------------ y /------------ y /------------ y /------------

%e x 1 2 3 4 5 6 x 1 2 3 4 5 6 x 1 2 3 4 5 6 x 1 2 3 4 5 6

%e .

%e 17-20-5-2 20-17-5-2 17-13-8-2 17-8-10-5

%e 9 X 25 9 X 26 10 X 25 13 X 26

%e .

%e 4 | . . . . X . 4 | . . X . . . 4 | . . X . . .

%e 3 | . . . . . . 3 | . . . . . . 3 | . . . . . X

%e 2 | X . . . . X 2 | X . . . . X 2 | X . . . . .

%e 1 | . X . . . . 1 | . . X . . . 1 | . . X . . .

%e y /------------ y /------------ y /------------

%e x 1 2 3 4 5 6 x 1 2 3 4 5 6 x 1 2 3 4 5 6

%e .

%e 17-5-20-2 10-13-8-5 13-10-8-5

%e 18 X 25 9 X 25 9 X 26

%e .

%o (PARI) see Pfoertner link.

%Y Cf. A353447 (diagonal), A353449, A353450, A353451, A353533, A354700.

%Y The general case without excluding the corners of the grid rectangle is covered in A354700 and A354701.

%K nonn,tabl

%O 3,5

%A _Hugo Pfoertner_ and _Rainer Rosenthal_, May 02 2022