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

1, 144, 112, 340, 696, 1468, 2528, 4388, 6760, 10444, 14928, 21364, 28952, 39260, 51136, 66628, 84168, 106348, 131120, 161684, 195448, 236284, 280992, 334180, 391976, 459788, 533008, 617908, 709080, 813724, 925568, 1052804, 1188232, 1341100, 1503216, 1684948

Offset: 1

Author

Omar Aceval Garcia, Dec 31 2021

Keywords

• nonn
• easy

Comments

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.

Examples

			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.
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).
		

Links

• Jinyuan Wang, Table of n, a(n) for n = 1..1000
• Omar Aceval Garcia, On the Number of Maximum Inner Distance Latin Squares, arXiv:2112.13912 [math.CO], 2021.
• Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Crossrefs

Cf. A005899, A069894, A248800.

Formula

a(n) = 4*n + ( n^2 + 3/2 + (1/2)*(-1)^n )^2 for n >= 3.

a(n) = 4*n + A248800(n)^2 for n >= 3.

For n >= 5, a(n) - a(n-2) = 8*n^3 - 24*n^2 + (44 + 4*(-1)^n)*n - 20 - 4*(-1)^n.

For n >= 7, a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) + (48 + 16*(-1)^n)*(n-2).

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

Extensions

More terms from Jinyuan Wang, Jan 01 2022