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A350453 Number of Latin squares of order 2n with maximum inner distance with fixed entry 1 in cell (1,1).

%I A350453 #31 Feb 04 2022 01:59:43
%S A350453 1,144,112,340,696,1468,2528,4388,6760,10444,14928,21364,28952,39260,
%T A350453 51136,66628,84168,106348,131120,161684,195448,236284,280992,334180,
%U A350453 391976,459788,533008,617908,709080,813724,925568,1052804,1188232,1341100,1503216,1684948
%N A350453 Number of Latin squares of order 2n with maximum inner distance with fixed entry 1 in cell (1,1).
%C A350453 The inner distance of a matrix with entries in [1,n] is the minimum of distances between vertically or horizontally adjacent entries. For example, every Latin square of order 2, 3, or 4 has inner distance 1, since there are consecutive integers which are adjacent. The distance between x and y in [1,n] with x < y is the minimum of y - x and n + x - y.
%H A350453 Jinyuan Wang, <a href="/A350453/b350453.txt">Table of n, a(n) for n = 1..1000</a>
%H A350453 Omar Aceval Garcia, <a href="https://arxiv.org/abs/2112.13912">On the Number of Maximum Inner Distance Latin Squares</a>, arXiv:2112.13912 [math.CO], 2021.
%H A350453 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-6,0,6,-2,-2,1).
%F A350453 a(n) = 4*n + ( n^2 + 3/2 + (1/2)*(-1)^n )^2 for n >= 3.
%F A350453 a(n) = 4*n + A248800(n)^2 for n >= 3.
%F A350453 For n >= 5, a(n) - a(n-2) = 8*n^3 - 24*n^2 + (44 + 4*(-1)^n)*n - 20 - 4*(-1)^n.
%F A350453 For n >= 7, a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) + (48 + 16*(-1)^n)*(n-2).
%F A350453 G.f.: x*(1 + 142*x - 178*x^2 - 166*x^3 + 656*x^4 + 62*x^5 - 622*x^6 + 190*x^7 + 207*x^8 - 100*x^9)/((1 - x)^5*(1 + x)). - _Stefano Spezia_, Jan 01 2022
%e A350453 For example there are 144 Latin squares of order 4 (with a 1 in the top left), all of which have maximum inner distance. There are only 112 such Latin squares of order 6, 340 of order 8, etc.
%e A350453 Every Latin square of order 4 by default has the maximum inner distance; the same is not true for any order higher than 4, which may explain why a(2) > a(3).
%Y A350453 Cf. A005899, A069894, A248800.
%K A350453 nonn,easy
%O A350453 1,2
%A A350453 _Omar Aceval Garcia_, Dec 31 2021
%E A350453 More terms from _Jinyuan Wang_, Jan 01 2022