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Number of main classes of diagonal Latin squares of order n that contain a doubly symmetric square.
4

%I #26 Oct 22 2023 12:38:38

%S 1,0,0,1,0,0,0,47,0,0,0

%N Number of main classes of diagonal Latin squares of order n that contain a doubly symmetric square.

%C A doubly symmetric square has symmetries in both the horizontal and vertical planes (see A292517).

%C Every doubly symmetric diagonal Latin square also has central symmetry. The converse is not true in general. It follows that a(n) <= A340545(n). - _Eduard I. Vatutin_, May 28 2021

%H Eduard I. Vatutin, <a href="https://vk.com/wall162891802_1470">On the number of main classes of one plane and double plane symmetric diagonal Latin squares of orders 1-11</a> (in Russian).

%H Eduard I. Vatutin, <a href="https://vk.com/wall162891802_1635">On the interconnection between double and central symmetries in diagonal Latin squares</a> (in Russian).

%H E. I. Vatutin, <a href="http://evatutin.narod.ru/evatutin_dls_spec_types_list.pdf">Special types of diagonal Latin squares</a>, Cloud and distributed computing systems in electronic control conference, within the National supercomputing forum (NSCF - 2022). Pereslavl-Zalessky, 2023. pp. 9-18. (in Russian)

%H <a href="/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>.

%e An example of a doubly symmetric diagonal Latin square:

%e 0 1 2 3 4 5 6 7

%e 3 2 7 6 1 0 5 4

%e 2 3 1 0 7 6 4 5

%e 6 7 5 4 3 2 0 1

%e 7 6 3 2 5 4 1 0

%e 4 5 0 1 6 7 2 3

%e 5 4 6 7 0 1 3 2

%e 1 0 4 5 2 3 7 6

%e In the horizontal direction there is a one-to-one correspondence between elements 0 and 7, 1 and 6, 2 and 5, 3 and 4.

%e In the vertical direction there is also a correspondence between elements 0 and 1, 2 and 4, 6 and 7, 3 and 5.

%Y Cf. A287764, A292517, A287650, A340545, A340546.

%K nonn,more,hard

%O 1,8

%A _Eduard I. Vatutin_, Jan 11 2021

%E Name clarified by _Andrew Howroyd_, Oct 22 2023