A355119 a(n) is the number of order-n magic triangles composed of the numbers from 1 to n(n+1)/2 in which the sum of the k-th row and the (n-k+1)-st row is the same for all k and all three directions, counted up to rotations and reflections.

1, 1, 0, 0, 7584, 5546793216

Offset: 1

Author

Donghwi Park, Jun 19 2022

Keywords

• nonn
• more

Comments

The magic sum is (n(n+1)/2 + 1)(n+1)/2.

For n >= 3, a(n) is a multiple of 6 because the rotation of only three corners does not affect the sum of the 1st row and n-th row.

This magic triangle is an analog of magic triangles from St. Olaf College, which are published in the Pi Mu Epsilon Journal (Fall 2021). Their magic triangles use square numbers of triangles.

Examples

			a(1) and a(2) are trivially 1.
a(3) is trivially 0 because the sum of 2nd row cannot be same for each direction.
a(4k) for positive integers k is trivially 0 because the magic sums are not integers in this cases.
An example of a solution at n=5:
         4
       7   9
     12  1  11
   14  2   3  13
  6  15  10  8  5
An example of a solution at n=6:
          9
        20 18
      21  8  13
    11   3  2  19
   10  6  4  7   12
 1  16  17 15  14  5
		

Links

• Gabriel Hale, Bjorn Vogen, and Matthew Wright, Magic Triangles, The Pi Mu Epsilon Journal (Fall 2021).
• Donghwi Park, Source code for a(5)
• Donghwi Park, Source code for a(6)

Crossrefs

Cf. A000217, A006052, A008586, A342467, A356643, A356808.

Formula

a(n) = 0 if n is a multiple of 4. - Stefano Spezia, Jun 20 2022

Extensions

a(6) from Donghwi Park, Dec 31 2023