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A356643 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.

%I A356643 #14 Oct 05 2022 05:00:24
%S A356643 1,0,0,0,612,22411,0
%N A356643 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 A356643 The magic sum is n*(n*(n + 1)/2 + 1)/2.
%H A356643 Donghwi Park, <a href="https://github.com/gwahak/mathematics/blob/master/A356643.ipynb">Source code for a(4)~a(6)</a>
%F A356643 a(n) = 0 if n is a multiple of 4 minus 1.
%e A356643 a(1) is trivially 1.
%e A356643 a(2) is trivially 0.
%e A356643 a(4k-1) for positive integers k is trivially 0 because the magic sums are not integers in these cases.
%e A356643 a(4) is 0.
%e A356643 An example of a solution at n=5:
%e A356643 .
%e A356643                  1
%e A356643              15     5
%e A356643            9     4     7
%e A356643        12     6     8    13
%e A356643      3    11     2    10    14
%e A356643 .
%e A356643 An example of a solution at n=6:
%e A356643 .
%e A356643                  5
%e A356643              19    16
%e A356643           12     1    20
%e A356643         9     6    10     8
%e A356643     18    11     7    21     2
%e A356643   3    17    13     4    14    15
%e A356643 .
%Y A356643 Cf. A000217 (triangular number), A006052 (magic square), A004767, A342467, A355119.
%K A356643 nonn,more
%O A356643 1,5
%A A356643 _Donghwi Park_, Aug 19 2022