A353449 -id:A353449 - cp's OEIS frontend

A353448 Number w is in this sequence if every frame w X h, w >= h >= 3, contains more distinct distance quadrilaterals with corners interior to the 4 sides with concurrent diagonals, i.e., both ascending or both descending, than non-concurrent diagonals, or equivalently A353450(w,h) >= A353449(w,h).

3, 4, 5, 7, 10, 11, 13, 14, 15, 16, 17, 19, 21, 23, 25, 29, 31, 33, 34, 37, 41, 43, 45, 46, 49, 55, 57, 61, 67, 73, 79, 81, 85, 89, 91, 97, 109, 113, 121, 127, 133, 141, 145, 151, 157, 161, 169, 181, 193, 201, 205, 209, 211, 217, 221, 225, 241, 253, 261, 265, 271

Offset: 1

Rainer Rosenthal, May 22 2022

nonn

It is conjectured that this sequence is a subsequence A160007 except for the small terms <= 46 (verified for all w < 661). The example section depicts the way this sequence matches the irregular pattern of A160007. Numbers w > 100 were computed by Hugo Pfoertner.

w = 617 is one of the rare occurrences of remainder 17 mod 60 in A160007, and it is also in this sequence. One might suspect that there would be less and less hits, but the time-consuming computation successfully countered the intuition.

			.
w = 5 is in this sequence:
.
                        5 | . . . C .
   4 | . . C . .        4 | .       .    w = 5 is in this sequence because all
   3 | .       B        3 | .       B    quadrilaterals in (5,4) and (5,5)
   2 | D       .        2 | D       .    shown in the example section of A353450
   1 | . A . . .        1 | . A . . .    have concurrent diagonals.
   y /----------        y /----------
     x 1 2 3 4 5          x 1 2 3 4 5
.
w = 6 is *not* in this sequence:
.
   4 | . . . C . .         w = 6 is not in this sequence because of the single
   3 | D         .         quadrilateral in (6,4) shown in the example section
   2 | .         B         of A353449. Diagonal AC is rising while diagonal DB
   1 | . A . . . .         is falling (non-concurrent diagonals).
   y /------------         There is no (6,4) quadrilateral with all distances
     x 1 2 3 4 5 6         distinct and with concurrent diagonals!
.
            123456789012345678901234567890123456789012345678901234567890
    1 -  60   ::: x  :: x:::: x x : x   x x ::  x   x x ::  x     x x
   61 - 120 x     x   . x     x x   x   x x     x   . . .   x   x .
  121 - 180 x     x     x     . x   x     x     x   x .     x     . .
  181 - 240 x     .     x   . . x   x   x x     x   x . x   .     .
  241 - 300 x     .     x     . x   x     x .   .   x .     x     .
  301 - 360 x   . . .   x     . x   x     .     x   . .     . . . .
  361 - 420 x   . . .   .     x .   x     x .   x   x .     x     . .
  421 - 480 x     .     x     . x   .   . .     x   . x .   x     . .
  481 - 540 x     .   . .   . . .   x     .     .   x .     x   . .
  541 - 600 x   . .   . .     . x   .     . .   x   . .     x     .
  601 - 660 x     . .   x   x . .   x     x     .   x . .   x .   .
.
Legend:
  "x" marks numbers w belonging to this sequence and to A160007.
  ":" marks numbers w belonging to this sequence only.
  "." marks numbers w belonging to A160007 only.

Rainer Rosenthal, Table of n, a(n) for n = 1..96
Hugo Pfoertner, Comparison to A225730, using Plot 2.

Cf. A160007, A353532 ("all tetrapods"), A353449 ("unisense"), A353450 ("contrasense").

Cf. A225730 (has many terms in common when 1 is added, see also comparison plot).

A353532 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.

Original entry on oeis.org

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, 198, 301, 190, 10, 49, 103, 192, 302, 444, 583, 379, 10, 58, 127, 244, 387, 576, 754, 1039, 616, 18, 84, 180, 329, 509, 756, 989, 1334, 1680, 987, 18, 94, 209, 389, 611, 910, 1203, 1618, 2052, 2581, 1426

Offset: 3

Hugo Pfoertner and Rainer Rosenthal, May 02 2022

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

			The triangle begins
    \ m 3   4    5    6    7    8    9   10
   n \-------------------------------------
   3 |  0,  |    |    |    |    |    |    |
   4 |  0,  0,   |    |    |    |    |    |
   5 |  0,  3,   1,   |    |    |    |    |
   6 |  1,  7,  12,  11,   |    |    |    |
   7 |  1, 11,  26,  52,  40,   |    |    |
   8 |  4, 23,  50,  94, 147, 105,   |    |
   9 |  4, 30,  69, 127, 198, 301, 190,   |
  10 | 10, 49, 103, 192, 302, 444, 583, 379
.
.
   4 | . C . . .    There are six squared distances.
   3 | . . . . .    They are arranged as follows:
   2 | D . . . B      AB-BC-CD-DA  (counterclockwise)
   1 | . A . . .      AC X DB      (across)
   y /----------    Here: AB = 3^2 + 1^2 = 10,
     x 1 2 3 4 5          BC = 13, CD = 5, DA = 2,
.                         AC =  9, DB = 16
      10-13-5-2  <==== yielding this
      9 X 16     <==== description
.
.
T(5,4) = a(5) = 3:
.
   4 | . X . . .     4 | . X . . .     4 | . . X . .
   3 | . . . . .     3 | . . . . X     3 | . . . . X
   2 | X . . . X     2 | X . . . .     2 | X . . . .
   1 | . X . . .     1 | . X . . .     1 | . X . . .
   y /----------     y /----------     y /----------
     x 1 2 3 4 5       x 1 2 3 4 5       x 1 2 3 4 5
.
      10-13-5-2          13-10-5-2          13-5-8-2
      9 X 16             9 X 17             10 X 17
.
T(5,5) = a(6) = A353447(5) = 1:
.
   5 | . . . X .
   4 | . . . . .
   3 | . . . . X    13-5-18-2
   2 | X . . . .    20 X 17
   1 | . X . . .
   y /----------
     x 1 2 3 4 5
.
T(6,3) = a(7) = 1:
.
   3 | . . . X . .
   2 | X . . . . X    17-5-10-2
   1 | . X . . . .    8 X 25
   y /------------
     x 1 2 3 4 5 6
.
T(6,4) = a(8) = 7:
.
   4 | . X . . . .   4 | . X . . . .   4 | . . X . . .   4 | . . . X . .
   3 | . . . . . .   3 | . . . . . X   3 | . . . . . .   3 | X . . . . .
   2 | X . . . . X   2 | X . . . . .   2 | X . . . . X   2 | . . . . . X
   1 | . X . . . .   1 | . X . . . .   1 | . X . . . .   1 | . X . . . .
   y /------------   y /------------   y /------------   y /------------
     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
.
       17-20-5-2         20-17-5-2         17-13-8-2         17-8-10-5
       9 X 25            9 X 26            10 X 25           13 X 26
.
   4 | . . . . X .   4 | . . X . . .   4 | . . X . . .
   3 | . . . . . .   3 | . . . . . .   3 | . . . . . X
   2 | X . . . . X   2 | X . . . . X   2 | X . . . . .
   1 | . X . . . .   1 | . . X . . .   1 | . . X . . .
   y /------------   y /------------   y /------------
     x 1 2 3 4 5 6     x 1 2 3 4 5 6     x 1 2 3 4 5 6
.
       17-5-20-2         10-13-8-5         13-10-8-5
       18 X 25           9 X 25            9 X 26
.

Rainer Rosenthal, Rows n = 3..100, flattened
Hugo Pfoertner, PARI program

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

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

PARI
```
see Pfoertner link.
```

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.

Original entry on oeis.org

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

Offset: 3

Rainer Rosenthal, May 13 2022

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.

			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.

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

Cf. A353532 ("all"), A353449 ("unisense"), A353451 ("static").

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.

Original entry on oeis.org

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

Rainer Rosenthal, May 13 2022

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.

			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.

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

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

cp's OEIS Frontend

A353448 Number w is in this sequence if every frame w X h, w >= h >= 3, contains more distinct distance quadrilaterals with corners interior to the 4 sides with concurrent diagonals, i.e., both ascending or both descending, than non-concurrent diagonals, or equivalently A353450(w,h) >= A353449(w,h).

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A353532 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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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.

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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.

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