OFFSET
3,5
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.
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, 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.
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Rainer Rosenthal, May 13 2022
STATUS
approved