The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A353450 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. 5
0, 0, 0, 0, 1, 1, 0, 0, 4, 5, 0, 2, 8, 18, 16, 0, 5, 14, 28, 50, 36, 0, 7, 23, 42, 75, 109, 77, 0, 10, 34, 65, 110, 157, 223, 143, 0, 14, 45, 89, 151, 223, 314, 423, 265, 0, 18, 58, 116, 186, 274, 386, 519, 684, 400, 0, 23, 73, 145, 239, 355, 491, 652, 870, 1069, 622 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
3,9
COMMENTS
Property "(x3-x1)*(y4-y2) < 0" holds iff the diagonals (spokes) of the quadrilateral are concurrent, i.e., their slopes are both positive or both negative. In this case the spokes are tilted in a different sense: clockwise and counterclockwise (see example). The framed quadrilateral may be classified as "contrasense" 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.
LINKS
Rainer Rosenthal, Rows n = 3..100, flattened
EXAMPLE
The triangle begins
\ m 3 4 5 6 7 8 9 10
n \-------------------------------------
3 | 0 | | | | | | |
4 | 0, 0 | | | | | |
5 | 0, 1, 1 | | | | |
6 | 0, 0, 4, 5 | | | |
7 | 0, 2, 8, 18, 16 | | |
8 | 0, 5, 14, 28, 50, 36 | |
9 | 0, 7, 23, 42, 75, 109, 77 |
10 | 0, 10, 34, 65, 110, 157, 223, 143
.
T(5,4) = 1 because of the third example for (5,4) in A353532.
.
4 | . . C . .
3 | . . . . B A = (x1,1) = (2,1), B = (5,y2) = (5,3)
2 | D . . . . C = (x3,4) = (3,4), D = (1,y4) = (1,2)
1 | . A . . .
y /---------- (x3-x1) * (y4-y2) = (3-2)*(2-3) < 0
x 1 2 3 4 5
.
Spokes AC and BD are tilted in different directions ("contrasense"). AC has positive slope and is tilted to the right (clockwise), DB also has same sign of slope, but is tilted to the left (counterclockwise).
.
T(5,5) = 1 because of the third example for (5,5) in A353532.
.
5 | . . . C .
4 | . . . . . A = (x1,1) = (2,1), B = (5,y2) = (5,3)
3 | . . . . B C = (x3,5) = (4,5), D = (1,y4) = (1,2)
2 | D . . . .
1 | . A . . . (x3-x1) * (y4-y2) = (4-2)*(2-3) < 0
y /----------
x 1 2 3 4 5
.
Spokes AC and DB are tilted in different directions ("contrasense") like in the example before.
CROSSREFS
Cf. A353532 ("all"), A353449 ("unisense"), A353451 ("static").
Sequence in context: A199390 A270914 A048837 * A087492 A366362 A199967
KEYWORD
nonn,tabl
AUTHOR
Rainer Rosenthal, May 13 2022
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 23 10:34 EDT 2024. Contains 372760 sequences. (Running on oeis4.)