Klaus Nagel - cp's OEIS frontend

User: Klaus Nagel

Klaus Nagel has authored 2 sequences.

A371969 Perimeters of triangles with integer sides, which can be decomposed into 3 triangles that have a common vertex strictly inside the surrounding triangle and also integer sides.

49, 50, 54, 64, 75, 78, 80, 88, 90, 91, 98, 100, 104, 108, 112, 117, 120, 121, 125, 126, 128, 133, 136, 140, 144, 147, 150, 156, 160, 162, 165, 168, 169, 170, 175, 176, 180, 182, 184, 188, 192, 195, 196, 198, 200, 203, 208, 210, 216, 220, 224, 225, 231, 234, 238, 240

Offset: 1

Klaus Nagel and Hugo Pfoertner, Apr 14 2024

nonn

			a(1) = 49 is the perimeter of the first decomposable triangle with sides of the outer triangle [8, 19, 22], and sides meeting at the 4th "inner" vertex: 17, 6, 4. The 3 inner triangles have sides [8, 4, 6], [19, 17, 4], and [22, 6, 17].

These triangles can be viewed as degenerate tetrahedrons, in which all triangular inequalities for the faces are satisfied, and the Cayley-Menger determinant, which determines whether the 4th vertex yields a valid tetrahedron, takes the value 0.

Klaus Nagel, Illustration of a(1) = 49.
Klaus Nagel, Illustration of a(2) = 50.
Klaus Nagel, Illustration of a(3) = 54.
Klaus Nagel, Illustration of a(4) = 64; another triangle is [20, 21, 23].
Hugo Pfoertner, List of triangles up to perimeter 400; one example of each.
Wikipedia, Cayley-Menger determinant

Cf. A070083, A316841, A342720, A371345, A371973.

H(a,b,c) = {my (s=(a+b+c)/2); s*(s-a)*(s-b)*(s-c)};
CM(w1,w2,w3,v1,v2,v3) = matdet([0,1,1,1,1; 1,0,w3^2,w2^2,v1^2; 1,w3^2,0,w1^2,v2^2; 1,w2^2,w1^2,0,v3^2; 1,v1^2,v2^2,v3^2,0]);
is_a371969(peri) = {forpart (w=peri, my (A=H(w[1],w[2],w[3]), epsA=1e-12); for (v1=1, w[3]-2, for (v2=w[3]-v1+1, w[3]-2, my (A3=H(w[3],v2,v1)); if (A3>=A, next); for (v3=1, w[3]-2, if (v3+v2<=w[1] || v3+v1<=w[2], next); my (A1=H(w[1],v2,v3)); if (A1>=A, next); my (A2=H(w[2],v1,v3)); if (A2>=A, next); my (C=CM(w[1],w[2],w[3],v1,v2,v3)); if (C==0 && abs(sqrt(A)-sqrt(A1)-sqrt(A2)-sqrt(A3)) < epsA,
\\ print (peri," ",Vec(w)," ",[v1,v2,v3]);
return(1))))), [1,(peri-1)\2], [3,3]); 0};
for (n=3, 100, if (is_a371969(n), print1(n,", ")))

A367147 Index of matching grid point in the bijection between two infinite triangular grids with one grid rotated by Pi/6 around the common point (0,0), using an enumeration of the grid points by A307014 and A307016.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 12, 14, 15, 9, 17, 18, 29, 7, 8, 23, 10, 11, 30, 13, 20, 21, 22, 33, 24, 16, 26, 27, 28, 36, 42, 19, 38, 39, 25, 41, 31, 32, 57, 34, 35, 60, 54, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 72, 37, 63, 66, 40, 69, 55, 73, 74, 56, 76, 77, 58, 79, 80, 59

Offset: 0

Klaus Nagel and Hugo Pfoertner, Nov 06 2023

nonn

The methods used to achieve a distance-limited bijection of the points of two square grids (see A307110) are applied here to triangular grids. The two grids, which are rotated by 30 degrees = Pi/6 from each other, are assigned the colors red and blue to distinguish them, which are also used in the illustrations. The blue triangular grid is turned clockwise by 15 degrees = Pi/12, all points are lined up on parallel lines with inclination Pi/12 towards the vertical axis. These are called blue lines. The vertical distance between adjacent points is cos(Pi/12). The same is done for the red grid with a CCW rotation of Pi/12. The whole plane is divided into stripes with a width of cos(Pi/12) ~= 0.9659. Every blue line and every red line contains exactly one grid point of its color in each stripe. The blue and red lines alternately intersect the horizontal centerline of a stripe. The distance between two intersections of the same color is d = sqrt(3)/(2*cos(Pi/12)). The bijection maps the section of a blue line in a stripe to the section of the unique red line, that intersects the centerline less than d/2 away. The grid points on these two line sections are the partners of the tile bijection.
While the method described only finds a minimum of the maximum distance of approximately 0.9659 by assigning the bijection partners using tiles, applying the Hopcroft-Karp algorithm to the bipartite graph corresponding to a sufficiently large section of the two infinite grids achieves significantly lower maximum distances. We conjecture that an upper bound for the maximum distance is sqrt(2)/2~=0.7071. See the corresponding link.
A method that reduces the maximal occurring bijection distance to its conjectured minimum, and only requires local rearrangements, as described for the square grids in A307731, is currently not known in the present case of the triangular grids.

			   n  A307014(n)        Bijection partner
   |  |  A307016(n)     in rotated grid
   |  |  |                          rotated by Pi/6
   |  |  |   x    y     i  j  a(n)   u      v   Distance([x,y],[u,v])
   0  0  0  0.0  0.0    0  0   0    0.0    0.0  0.0
   1  1  0  1.0  0.0    1  0   1    0.866  0.5  0.51764
   2  0  1  0.5  0.866  0  1   2    0.0    1.0  0.51764
   3 -1  1 -0.5  0.866 -1  1   3   -0.866  0.5  0.51764
   4 -1  0 -1.0  0.0   -1  0   4   -0.866 -0.5  0.51764
   5  0 -1 -0.5 -0.866  0 -1   5    0.0   -1.0  0.51764
   6  1 -1  0.5 -0.866  1 -1   6    0.866 -0.5  0.51764
   7  1  1  1.5  0.866  2 -1  12    1.732  0.0  0.89658
   8 -1  2  0.0  1.732  0  2  14    0.0    2.0  0.26795
   9 -2  1 -1.5  0.866 -2  2  15   -1.732  1.0  0.26795
  10 -1 -1 -1.5 -0.866 -2  1   9   -1.732  0.0  0.89658
  11  1 -2  0.0 -1.732  0 -2  17    0.0   -2.0  0.26795
  12  2 -1  1.5 -0.866  2 -2  18    1.732 -1.0  0.26795
  13  2  0  2.0  0.0    3 -2  29    2.598 -0.5  0.77955
  14  0  2  1.0  1.732  1  1   7    0.866  1.5  0.26795
  15 -2  2 -1.0  1.732 -1  2   8   -0.866  1.5  0.26795

Klaus Nagel, Stripes and tiles used for mapping.
Hugo Pfoertner, Numbering of points in red and blue grid.
Hugo Pfoertner, Notes on the minimum achievable bijection distance, Nov 2023.
Hugo Pfoertner, PARI program.
Wikipedia, Hopcroft-Karp algorithm

Cf. A307014, A307016, A307110, A307731, A367148.

\\ See linked file; function call to output data:
a367147(70)

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User: Klaus Nagel

A371969 Perimeters of triangles with integer sides, which can be decomposed into 3 triangles that have a common vertex strictly inside the surrounding triangle and also integer sides.

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