A355119 - OEIS

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.

%I #38 Jan 30 2024 16:05:18

%S 1,1,0,0,7584,5546793216

%N 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.

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

%C 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.

%C 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.

%H Gabriel Hale, Bjorn Vogen, and Matthew Wright, <a href="https://www.mlwright.org/docs/magic_triangles.pdf">Magic Triangles</a>, The Pi Mu Epsilon Journal (Fall 2021).

%H Donghwi Park, <a href="https://github.com/gwahak/mathematics/blob/master/A355119.ipynb">Source code for a(5)</a>

%H Donghwi Park, <a href="https://github.com/gwahak/mathematics/blob/master/A355119-a(6).ipynb">Source code for a(6)</a>

%F a(n) = 0 if n is a multiple of 4. - _Stefano Spezia_, Jun 20 2022

%e a(1) and a(2) are trivially 1.

%e a(3) is trivially 0 because the sum of 2nd row cannot be same for each direction.

%e a(4k) for positive integers k is trivially 0 because the magic sums are not integers in this cases.

%e An example of a solution at n=5:

%e 4

%e 7 9

%e 12 1 11

%e 14 2 3 13

%e 6 15 10 8 5

%e An example of a solution at n=6:

%e 9

%e 20 18

%e 21 8 13

%e 11 3 2 19

%e 10 6 4 7 12

%e 1 16 17 15 14 5

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

%K nonn,more

%O 1,5

%A _Donghwi Park_, Jun 19 2022

%E a(6) from _Donghwi Park_, Dec 31 2023