A356643 - OEIS

%I #14 Oct 05 2022 05:00:24

%S 1,0,0,0,612,22411,0

%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)-th row is same for all k and all three directions, counted up to rotations and reflections.

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

%H Donghwi Park, <a href="https://github.com/gwahak/mathematics/blob/master/A356643.ipynb">Source code for a(4)~a(6)</a>

%F a(n) = 0 if n is a multiple of 4 minus 1.

%e a(1) is trivially 1.

%e a(2) is trivially 0.

%e a(4k-1) for positive integers k is trivially 0 because the magic sums are not integers in these cases.

%e a(4) is 0.

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

%e .

%e                  1

%e              15     5

%e            9     4     7

%e        12     6     8    13

%e      3    11     2    10    14

%e .

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

%e .

%e                  5

%e              19    16

%e           12     1    20

%e         9     6    10     8

%e     18    11     7    21     2

%e   3    17    13     4    14    15

%e .

%Y Cf. A000217 (triangular number), A006052 (magic square), A004767, A342467, A355119.

%K nonn,more

%O 1,5

%A _Donghwi Park_, Aug 19 2022