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A307171 Maximum number of partial loops in a diagonal Latin square of order n.

%I A307171 #29 Jun 13 2021 03:46:03
%S A307171 0,0,0,12,8,21,53,112
%N A307171 Maximum number of partial loops in a diagonal Latin square of order n.
%C A307171 A loop in a Latin square is a sequence of cells v1=L[i1,j1] -> v2=L[i1,j2] -> v1=L[i2,j2] -> ... -> v2=L[im,j1] -> v1=L[i1,j1] of length 2*m that consists of a pair of values {v1, v2}. A partial loop is a loop of length < 2*n.
%H A307171 E. I. Vatutin, <a href="http://forum.boinc.ru/default.aspx?g=posts&amp;m=92687#post92687">Discussion about properties of diagonal Latin squares at forum.boinc.ru</a> (in Russian).
%H A307171 E. I. Vatutin, <a href="https://vk.com/wall162891802_1321">About the minimum and maximum number of partial loops in a diagonal Latin squares of order 8</a> (in Russian).
%H A307171 Eduard Vatutin, Alexey Belyshev, Natalia Nikitina, and Maxim Manzuk, <a href="https://doi.org/10.1007/978-3-030-66895-2_9">Evaluation of Efficiency of Using Simple Transformations When Searching for Orthogonal Diagonal Latin Squares of Order 10</a>, High-Performance Computing Systems and Technologies in Sci. Res., Automation of Control and Production (HPCST 2020), Communications in Comp. and Inf. Sci. book series (CCIS, Vol. 1304) Springer (2020), 127-146.
%H A307171 Eduard I. Vatutin, <a href="/A307171/a307171.txt">Proving list (best known examples)</a>.
%H A307171 <a href="/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>.
%e A307171 For example, the square
%e A307171   2 4 3 5 0 1
%e A307171   1 0 4 3 2 5
%e A307171   0 2 5 4 1 3
%e A307171   5 3 0 1 4 2
%e A307171   4 5 1 2 3 0
%e A307171   3 1 2 0 5 4
%e A307171 has a loop
%e A307171   2 4 . . . .
%e A307171   . . . . . .
%e A307171   . 2 . 4 . .
%e A307171   . . . . . .
%e A307171   4 . . 2 . .
%e A307171   . . . . . .
%e A307171 consisting of the sequence of cells L[1,1]=2 -> L[1,2]=4 -> L[3,2]=2 -> L[3,4]=4 -> L[5,4]=2 -> L[5,1]=4 -> L[1,1]=2 with length 6 < 12.
%e A307171 The total number of loops for this square is 21, all of which are partial.
%Y A307171 Cf. A307167, A307170.
%K A307171 nonn,more,hard
%O A307171 1,4
%A A307171 _Eduard I. Vatutin_, Mar 27 2019
%E A307171 a(8) added by _Eduard I. Vatutin_, Oct 06 2020