This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A307166 #34 Jun 02 2021 23:01:29
%S A307166 1,0,0,12,10,27,21,40
%N A307166 Minimum number of loops in a diagonal Latin square of order n.
%C A307166 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}.
%C A307166 For diagonal Latin squares of order 4 all loops are intercalates. - _Eduard I. Vatutin_, Oct 05 2020
%C A307166 From _Eduard I. Vatutin_, Oct 26 2020: (Start)
%C A307166 Every intercalate is a partial loop and every partial loop is a loop, so 0 <= A307163(n) <= A307170(n) <= a(n).
%C A307166 0 <= a(n) <= A307167(n).
%C A307166 (End)
%H A307166 E. I. Vatutin, <a href="http://forum.boinc.ru/default.aspx?g=posts&m=92687#post92687">Discussion about properties of diagonal Latin squares at forum.boinc.ru</a> (in Russian).
%H A307166 E. I. Vatutin, <a href="https://vk.com/wall162891802_1320">About the minimum and maximum number of loops in a diagonal Latin squares of order 8</a> (in Russian).
%H A307166 E. I. Vatutin, <a href="https://vk.com/wall162891802_1403">On the inequalities of the minimum and maximum numerical characteristics of diagonal Latin squares for intercalates, loops and partial loops</a> (in Russian).
%H A307166 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 A307166 Eduard I. Vatutin, <a href="/A307166/a307166.txt">Proving list (best known examples)</a>.
%H A307166 <a href="/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>.
%e A307166 For example, the square
%e A307166 2 4 3 5 0 1
%e A307166 1 0 4 3 2 5
%e A307166 0 2 5 4 1 3
%e A307166 5 3 0 1 4 2
%e A307166 4 5 1 2 3 0
%e A307166 3 1 2 0 5 4
%e A307166 has a loop
%e A307166 2 4 . . . .
%e A307166 . . . . . .
%e A307166 . 2 . 4 . .
%e A307166 . . . . . .
%e A307166 4 . . 2 . .
%e A307166 . . . . . .
%e A307166 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.
%e A307166 The total number of loops for this square is 21.
%Y A307166 Cf. A274806, A307163, A307164, A307167, A307170.
%K A307166 nonn,more,hard
%O A307166 1,4
%A A307166 _Eduard I. Vatutin_, Mar 27 2019
%E A307166 a(8) added by _Eduard I. Vatutin_, Oct 05 2020