A186434 -id:A186434 - cp's OEIS frontend

A271915 Number of ways to choose three distinct points from a 5 X n grid so that they form an isosceles triangle.

0, 24, 108, 248, 444, 672, 932, 1204, 1512, 1836, 2188, 2548, 2936, 3332, 3756, 4192, 4656, 5128, 5628, 6136, 6672, 7216, 7788, 8368, 8976, 9592, 10236, 10888, 11568, 12256, 12972, 13696, 14448, 15208, 15996, 16792

Offset: 1

N. J. A. Sloane, Apr 24 2016

nonn

Chai Wah Wu, Counting the number of isosceles triangles in rectangular regular grids, arXiv:1605.00180 [math.CO], 2016.
Index entries for linear recurrences with constant coefficients, signature (2, 0, -2, 1).

Row 5 of A271910.

Cf. A186434, A187452.

Join[{0, 24, 108, 248, 444, 672, 932, 1204, 1512, 1836, 2188, 2548, 2936, 3332}, LinearRecurrence[{2, 0, -2, 1}, {3756, 4192, 4656, 5128}, 20]] (* Jean-François Alcover, Sep 03 2018 *)

Conjectured g.f.: 4*x* (x^16-x^14+2*x^10+2*x^9-x^8-x^7 + 5*x^6+6*x^5+6*x^4+x^3-8*x^2-15*x-6) /((x+1)*(x-1)^3).
Conjectured recurrence: a(n) = 2*a(n-1)-2*a(n-3)+a(n-4) for n > 18.
The conjectured g.f. and recurrence are true. See paper in links. - Chai Wah Wu, May 07 2016

A189415 Number of trapezoids on an n X n grid (or geoboard).

Original entry on oeis.org

0, 1, 50, 490, 2618, 9519, 28432, 70796, 157912, 321161, 610482, 1082570, 1848362, 3003015, 4716792, 7204604, 10730528, 15530189, 22093410, 30723078, 42146178, 56981411, 75952240, 99685104, 129757248

Offset: 1

Martin Renner, Apr 21 2011

nonn

Eric Weisstein's World of Mathematics, Trapezoid.
Nathaniel Johnston, C program for computing terms.

Cf. A181945, A186434, A189413, A189414, A189416, A189418.

a(6)-a(25) from Nathaniel Johnston, Apr 25 2011

A279414 a(n) is the total number of isosceles triangles having a bounding box n X k where k is in the range 1 to n inclusive.

Original entry on oeis.org

0, 4, 14, 22, 36, 48, 58, 66, 104, 100, 110, 118, 164, 148, 174, 174, 232, 200, 266, 226, 300, 272, 290, 282, 412, 332, 362, 358, 440, 376, 494, 386, 572, 464, 490, 490, 660, 476, 546, 562, 756, 552, 718, 582, 760, 692, 682, 634, 1004, 716, 862, 746, 900, 744

Offset: 1

Lars Blomberg, Feb 16 2017

nonn

Row sums of A279413.

Lars Blomberg, Table of n, a(n) for n = 1..10000

Cf. A186434, A187452, A271910-A271913, A271915, A279413.

A190312 Number of scalene triangles on an n X n grid (or geoboard).

Original entry on oeis.org

0, 0, 40, 368, 1704, 5704, 15400, 36096, 75632, 145968, 263592, 451392, 738360, 1163552, 1774840, 2632344, 3808992, 5394752, 7493936, 10233832, 13759008, 18241312, 23877984, 30896984, 39551456, 50137240, 62983128, 78459880

Offset: 1

Martin Renner, May 08 2011

nonn

Eric Weisstein's World of Mathematics, Geoboard.
Eric Weisstein's World of Mathematics, Scalene Triangle.

Cf. A045996, A186434.

q[n_] :=
  Module[{sqDist, t0, t1, t2},
   (* Squared distances *)
   sqDist = {p_, q_} :> (Floor[p/n] - Floor[q/n])^2 + (Mod[p, n] - Mod[q, n])^2;
   (* Triads of points *)
   t0 = Subsets[Range[0, n^2 - 1], {3, 3}];
   (* Exclude collinear vertices *)
   t1 = Select[t0, Det[Map[{Floor[#/n], Mod[#, n], 1} &, {#[[1]], #[[2]], #[[
           3]]}]] != 0 &];
   (* Calculate sides *)
   t2 = Map[{#,
    Sort[{{#[[2]], #[[3]]}, {#[[3]], #[[1]]}, {#[[1]], #[[2]]}} /. sqDist]}&, t1];
   (* Select scalenes *)
   t2 = Select[t2,
      #[[2, 1]] != #[[2, 2]] && #[[2, 2]] != #[[2, 3]] && #[[2,3]] != #[[2, 1]] &];
   Return[Length[t2]];
   ];
Map[q[#] &, Range[9]] (* César Eliud Lozada, Mar 26 2021 *)

a(n) = A045996(n) - A186434(n).

A271911 Number of ways to choose three distinct points from a 2 X n grid so that they form an isosceles triangle.

Original entry on oeis.org

0, 4, 10, 16, 24, 32, 42, 52, 64, 76, 90, 104, 120, 136, 154, 172, 192, 212, 234, 256, 280, 304, 330, 356, 384, 412, 442, 472, 504, 536, 570, 604, 640, 676, 714, 752, 792, 832, 874, 916, 960, 1004, 1050, 1096, 1144, 1192, 1242, 1292, 1344, 1396, 1450, 1504

Offset: 1

N. J. A. Sloane, Apr 24 2016

			n=3: Label the points
1 2 3
4 5 6
There are 8 small isosceles triangles like 124 plus 135 and 246, so a(3) = 10.

Chai Wah Wu, Counting the number of isosceles triangles in rectangular regular grids, arXiv:1605.00180 [math.CO], 2016.
Index entries for linear recurrences with constant coefficients, signature (2, 0, -2, 1).

Row 2 of A271910.
Cf. A186434, A187452.
Same start as, but totally different from, 2*A213707.

LinearRecurrence[{2,0,-2,1},{0,4,10,16},60] (* Harvey P. Dale, May 10 2018 *)

Conjectured g.f.: 2*x*(2*x^2-x-2)/((x+1)*(x-1)^3). It would be nice to have a proof!
Conjectures from Colin Barker, Apr 24 2016: (Start)
a(n) = (-1+(-1)^n+16*n+2*n^2)/4, or equivalently, a(n) = (n^2+8*n)/2 if n even, (n^2+8*n-1)/2 if n odd.
a(n) = 2*a(n-1)-2*a(n-3)+a(n-4) for n>4. (End)
The conjectured g.f. and recurrence are true. See paper in links. - Chai Wah Wu, May 07 2016
a(n) = round(n*(n/2+3)) - 4. - Bill McEachen, Aug 10 2025

More terms from Harvey P. Dale, May 10 2018

A271912 Number of ways to choose three distinct points from a 3 X n grid so that they form an isosceles triangle.

Original entry on oeis.org

0, 10, 36, 68, 108, 150, 200, 252, 312, 374, 444, 516, 596, 678, 768, 860, 960, 1062, 1172, 1284, 1404, 1526, 1656, 1788, 1928, 2070, 2220, 2372, 2532, 2694, 2864, 3036, 3216, 3398, 3588, 3780, 3980, 4182, 4392, 4604, 4824, 5046, 5276, 5508, 5748, 5990, 6240, 6492, 6752, 7014, 7284, 7556

Offset: 1

N. J. A. Sloane, Apr 24 2016

nonn

			n=2: Label the points
1 2 3
4 5 6
There are 8 small isosceles triangles like 124 plus 135 and 246, so a(2) = 10.

Chai Wah Wu, Counting the number of isosceles triangles in rectangular regular grids, arXiv:1605.00180 [math.CO], 2016.
Index entries for linear recurrences with constant coefficients, signature (2, 0, -2, 1).

Row 3 of A271910.

Cf. A186434, A187452.

Join[{0, 10}, LinearRecurrence[{2, 0, -2, 1}, {36, 68, 108, 150}, 50]] (* Jean-François Alcover, Oct 10 2018 *)

Conjectured g.f.: 2*x*(2*x^4+4*x^3+2*x^2-8*x-5)/((x+1)*(x-1)^3).
Conjectured recurrence: a(n) = 2*a(n-1)-2*a(n-3)+a(n-4) for n > 6.
Conjectures from Colin Barker, Apr 25 2016: (Start)
a(n) = (-3*(47+(-1)^n)+64*n+10*n^2)/4 for n>2.
a(n) = (5*n^2+32*n-72)/2 for n>2 and even.
a(n) = (5*n^2+32*n-69)/2 for n>2 and odd.
(End)
The conjectured g.f. and recurrence are true. See paper in links. - Chai Wah Wu, May 07 2016

More terms from Jean-François Alcover, Oct 10 2018

A358532 a(n) is the row position of the next open point in the structure generated by adding the largest diamond possible at the next open point on a triangular grid of side n. See Comments and Example sections for more details.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 4, 1, 3, 7, 1, 3, 6, 4, 10, 1, 9, 4, 7, 9, 5, 14, 1, 11, 5, 7, 8, 11, 14, 19, 1, 6, 6, 24, 9, 14, 20, 1, 8, 8, 8, 20, 8, 19, 24, 30, 15, 19, 19, 19, 27, 1, 19, 15, 16, 20, 28, 8, 39, 11, 24, 1, 11, 16, 26, 28, 29, 30, 39, 50, 20, 31, 32, 33

Offset: 1

John Tyler Rascoe, Nov 20 2022

A structure of diamonds is built up successively by adding the largest possible diamond to the next open point within a triangular grid of side n. Each new diamond is added to the preceding structure of diamonds. At each step n, a new row of n open points is first added, extending the triangular grid.
Then the next open point is defined as the first open point encountered when the triangle is read by rows starting from the top row. a(n) is then the row position of the next open point.
Finally, starting at this open point the largest diamond that does not overlap any previous diamonds and fits within the triangular grid is added. Each diamond of side length k must cover exactly k^2 points, with the top corner on an open point. The points covered by the added diamond are then considered closed.
Is there a pattern for the values of n where a(n) = 1?

			Here zeros are the open points; closed points covered by the n-th diamond are replaced with n.
  ---------------------
  n=4       1          First a new row of 4 open points is added.
           2 3         Then the next open point is T(3,1) so a(4) = 1.
          4 0 0        Finally, the largest diamond fitting at T(3,1) is 1.
         0 0 0 0
  ---------------------
  n=5       1          First a new row of 5 open points is added.
           2 3         Then the next open point is T(3,2) so a(5) = 2.
          4 5 0        Finally, the largest diamond fitting at T(3,2) is 2.
         0 5 5 0
        0 0 5 0 0
  ---------------------
  n=6       1          First a new row of 6 open points is added.
           2 3         Then the next open point is T(3,3) so a(6) = 3.
          4 5 6        Finally, the largest diamond fitting at T(3,3) is 1.
         0 5 5 0
        0 0 5 0 0
       0 0 0 0 0 0

John Tyler Rascoe, Table of n, a(n) for n = 1..10000
John Tyler Rascoe, Python program

Cf. A045996, A186434, A194136, A239567, A279413.

Python
```
# see linked program
```

cp's OEIS Frontend

A271915 Number of ways to choose three distinct points from a 5 X n grid so that they form an isosceles triangle.

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A189415 Number of trapezoids on an n X n grid (or geoboard).

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A279414 a(n) is the total number of isosceles triangles having a bounding box n X k where k is in the range 1 to n inclusive.

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A190312 Number of scalene triangles on an n X n grid (or geoboard).

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Mathematica

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A271911 Number of ways to choose three distinct points from a 2 X n grid so that they form an isosceles triangle.

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Mathematica

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A271912 Number of ways to choose three distinct points from a 3 X n grid so that they form an isosceles triangle.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A358532 a(n) is the row position of the next open point in the structure generated by adding the largest diamond possible at the next open point on a triangular grid of side n. See Comments and Example sections for more details.

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Comments

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Python