cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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.

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

Views

Author

Hugo Pfoertner and Rainer Rosenthal, May 02 2022

Keywords

Comments

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

Examples

			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
.

Links

Crossrefs

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.

Programs

  • PARI
    see Pfoertner link.

A353447 a(n) is the number of tetrapods standing on the four edges of an n X n grid, so that no two feet are the same distance apart and no foot is on a corner. Tetrapods with congruent footprints are counted only once.

Original entry on oeis.org

0, 0, 1, 11, 40, 105, 190, 379, 616, 987, 1426, 2139, 2964, 4130, 5403, 7180, 9155, 11716, 14458, 18092, 22037, 26808, 31793, 38343, 45060, 53184, 61613, 71878, 82466, 95368, 108195, 123790, 140040, 158457, 177405, 200020, 223039, 248769, 275214, 306411, 337645
Offset: 3

Views

Author

Rainer Rosenthal, Apr 20 2022

Keywords

Comments

If we name the tetrapod's footprints "mini-frame", we can say that mini-frames span their grid, i.e., there is no smaller grid for them. Every corner-less set of points with distinct distances in a smallest possible n X n grid contains at least one mini-frame.

Examples

			  .
     . C .           a(3) = 0              . . . C .
     D . B   <===  since AB = CD           . . . . .
     . A .         is forbidden            . . . . B
                                           . . . . .
                        . C . .            D . . . .
      a(4) = 0  ===>    ? . . .            . A . . .
    (there is no        ? . . B         ______________
     space for D)       . A . .            a(5) = 1
                                     (No other solutions)
  .
    . . . . .           The tetrapod has 6 distinct
    D . . . .           squared distances 4, 5, 10,
    . . . . C   <=====  13, 17, 18, but it uses only
    . . . . .           three edges of the 5 X 5 grid.
    . A . B .           (Not allowed.)
  .

Links

Crossrefs

Cf. A193838, A271490, A335232, A351699, A351700, A353532.
The general case without excluding the corners of the grid rectangle is covered in A354700 and A354701.

Extensions

a(23) and beyond from Hugo Pfoertner, Apr 20 2022

A354700 T(w,h) is the number of non-congruent quadrilaterals whose vertices with integer coordinates (x_i, y_i) all lie on the perimeter of a rectangle of width w and height h, with no 3 points on the same edge of the rectangle, max(x_i) - min(x_i) = w and max(y_i) - min(y_i) = h, such that the 6 distances between the 4 vertices are distinct.

Original entry on oeis.org

0, 0, 0, 1, 4, 5, 2, 16, 36, 21, 8, 33, 69, 116, 71, 13, 52, 126, 201, 317, 181, 22, 84, 191, 299, 445, 639, 366, 28, 110, 249, 373, 581, 839, 1105, 585, 43, 157, 330, 529, 806, 1094, 1463, 1856, 1009, 50, 190, 407, 653, 1014, 1360, 1853, 2295, 2958, 1562
Offset: 1

Views

Author

Hugo Pfoertner, Jun 07 2022

Keywords

Comments

T(w,h) is a triangle read by rows, 1 <= h <= w.

Examples

			The triangle begins:
   0;
   0,   0;
   1,   4,   5;
   2,  16,  36,  21;
   8,  33,  69, 116,  71;
  13,  52, 126, 201, 317, 181;
  22,  84, 191, 299, 445, 639,  366;
  28, 110, 249, 373, 581, 839, 1105, 585
.
T(3,1) = 1:
  1 | D . . C  Squared distances:
  0 | A . B .  Sides: AB = 4, BC = 2, CD = 9, DA = 1;
  y /--------  Diagonals: AC = 10, BD = 5.
    x 0 1 2 3
.
T(3,2) = 4:
  2 | D . . C  Squared distances:
  1 | . . . .  Sides: AB = 1, BC = 8, CD = 9, DA = 4;
  0 | A B . .  Diagonals: AC = 13, BD = 5.
  y /--------
    x 0 1 2 3
  2 | . . . D  Squared distances:
  1 | . . . C  Sides: AB = 4, BC = 2, CD = 1, DA = 13;
  0 | A . B .  Diagonals: AC = 10, BD = 5.
  y /--------
    x 0 1 2 3
  2 | . . D .  Squared distances:
  1 | . . . C  Sides: AB = 9, BC = 1, CD = 2, DA = 8;
  0 | A . . B  Diagonals: AC = 10, BD = 5.
  y /--------
    x 0 1 2 3
  2 | . . C .  Squared distances:
  1 | D . . B  Sides: AB = 10, BC = 2, CD = 5, DA = 1;
  0 | A . . .  Diagonals: AC = 8, BD = 9.
  y /--------
    x 0 1 2 3
The last 2 quadrilaterals have the same set {1, 2, 5, 8, 9, 10} of squared distances, but with different allocation of sides and diagonals.
.
T(3,3) = 5:
  3 | . D . C    3 | . . . C    3 | . . . D    3 | . D . .    3 | . . D .
  2 | . . . .    2 | D . . .    2 | . . . .    2 | . . . C    2 | . . . .
  1 | . . . .    1 | . . . .    1 | . . . C    1 | . . . .    1 | . . . C
  0 | A B . .    0 | A B . .    0 | A B . .    0 | A B . .    0 | A . B .
  y /--------    y /--------    y /--------    y /--------    y /--------
    x 0 1 2 3      x 0 1 2 3      x 0 1 2 3      x 0 1 2 3      x 0 1 2 3
Quadrilaterals Q2 and Q3 have the same set {1, 4, 5, 10, 13, 18} of squared distances, but the allocation of sides and diagonals differ: Squared diagonals are AC, BD {18,5} in Q2, and {10,13} in Q3.

Links

Crossrefs

Cf. A353532, A354699, A354701 (diagonal).

Programs

  • PARI
    \\ See link.
Showing 1-3 of 3 results.

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