Some pairs of cyclic mutually orthogonal Hamiltonian double latin squares of order 2n. In each case, the given double latin square is orthogonal to the square with initial difference sequence (1,-2,3,...).

n=3:

{{-2,1,3,-1,2},{0,4,5,2,1,3}}

n=4:

none

n=5:

{{2,4,3,-1,5,1,-3,-4,-2},{0,2,6,9,8,3,4,1,7,5}}

{{4,-2,-1,-3,5,3,1,2,-4},{0,4,2,1,8,3,6,7,9,5}}

n=6:

none

n=7:

{{-2,-4,-6,1,3,5,7,-5,-3,-1,6,4,2},{0,12,8,2,3,6,11,4,13,10,9,1,5,7}}

{{2,4,6,3,-5,1,7,-1,5,-3,-6,-4,-2},{0,2,6,12,1,10,11,4,3,8,5,13,9,7}}

{{5,-4,1,-6,3,-2,7,2,-3,6,-1,4,-5},{0,5,1,2,10,13,11,4,6,3,9,8,12,7}}

{{-6,2,-4,-1,-3,-5,7,5,3,1,4,-2,6},{0,8,10,6,5,2,11,4,9,12,13,3,1,7}}

n=8:

{{-2,7,-3,1,4,-6,-5,8,5,6,-4,-1,3,-7,2},

{0,14,5,2,3,7,1,12,4,9,15,11,10,13,6,8}}

{{-2,7,-4,-6,-1,5,-3,8,3,-5,1,6,4,-7,2},

{0,14,5,1,11,10,15,12,4,7,2,3,9,13,6,8}}

{{-3,-2,4,7,-5,1,-6,8,6,-1,5,-7,-4,2,3},

{0,13,11,15,6,1,2,12,4,10,9,14,7,3,5,8}}

{{3,6,-4,1,-7,-5,2,8,-2,5,7,-1,4,-6,-3},

{0,3,9,5,6,15,10,12,4,2,7,14,13,1,11,8}}

{{5,1,3,6,4,7,2,8,-2,-7,-4,-6,-3,-1,-5},

{0,5,6,9,15,3,10,12,4,2,11,7,1,14,13,8}}

{{-5,3,-7,-2,-4,1,-6,8,6,-1,4,2,7,-3,5},

{0,11,14,7,5,1,2,12,4,10,9,13,15,6,3,8}}

{{6,1,4,2,5,7,3,8,-3,-7,-5,-2,-4,-1,-6},

{0,6,7,11,13,2,9,12,4,1,10,5,3,15,14,8}}

{{6,-3,-1,7,-4,2,5,8,-5,-2,4,-7,1,3,-6},

{0,6,3,2,9,5,7,12,4,15,13,1,10,11,14,8}}

n=9:

{{-1,4,7,-6,2,-8,-5,3,9,-3,5,8,-2,6,-7,-4,1},

{0,17,3,10,4,6,16,11,14,5,2,7,15,13,1,12,8,9}}

{{-1,7,-3,4,6,-2,8,-5,9,5,-8,2,-6,-4,3,-7,1},

{0,17,6,3,7,13,11,1,14,5,10,2,4,16,12,15,8,9}}

{{2,4,-5,-7,-8,3,1,6,9,-6,-1,-3,8,7,5,-4,-2},

{0,2,6,1,12,4,7,8,14,5,17,16,13,3,10,15,11,9}}

{{2,-5,-8,-4,1,6,7,-3,9,3,-7,-6,-1,4,8,5,-2},

{0,2,15,7,3,4,10,17,14,5,8,1,13,12,16,6,11,9}}

{{3,-2,6,-8,-4,7,-5,-1,9,1,5,-7,4,8,-6,2,-3},

{0,3,1,7,17,13,2,15,14,5,6,11,4,8,16,10,12,9}}

{{-3,-8,-6,1,2,4,-5,-7,9,7,5,-4,-2,-1,6,8,3},

{0,15,7,1,2,4,8,3,14,5,12,17,13,11,10,16,6,9}}

{{4,-1,7,-3,-5,6,-2,8,9,-8,2,-6,5,3,-7,1,-4},

{0,4,3,10,7,2,8,6,14,5,15,17,11,16,1,12,13,9}}

{{6,-2,-3,-8,5,1,4,-7,9,7,-4,-1,-5,8,3,2,-6},

{0,6,4,1,11,16,17,3,14,5,12,8,7,2,10,13,15,9}}

{{-6,-2,-3,-8,5,7,4,-1,9,1,-4,-7,-5,8,3,2,6},

{0,12,10,7,17,4,11,15,14,5,6,2,13,8,16,1,3,9}}

{{-6,4,1,-7,3,-2,-5,8,9,-8,5,2,-3,7,-1,-4,6},

{0,12,16,17,10,13,11,6,14,5,15,2,4,1,8,7,3,9}}

{{6,-8,3,1,-7,4,-5,2,9,-2,5,-4,7,-1,-3,8,-6},

{0,6,16,1,2,13,17,12,14,5,3,8,4,11,10,7,15,9}}

{{-7,1,3,4,-6,-5,8,-2,9,2,-8,5,6,-4,-3,-1,7},

{0,11,12,15,1,13,8,16,14,5,7,17,4,10,6,3,2,9}}

{{7,-1,4,3,-5,-6,-8,2,9,-2,8,6,5,-3,-4,1,-7},

{0,7,6,10,13,8,2,12,14,5,3,11,17,4,1,15,16,9}}

{{-7,4,-5,2,-8,-6,1,-3,9,3,-1,6,8,-2,5,-4,7},

{0,11,15,10,12,4,16,17,14,5,8,7,13,3,1,6,2,9}}

{{8,4,-2,5,1,-3,7,-6,9,6,-7,3,-1,-5,2,-4,-8},

{0,8,12,10,15,16,13,2,14,5,11,4,7,6,1,3,17,9}}

{{8,-5,-2,-4,1,6,7,3,9,-3,-7,-6,-1,4,2,5,-8},

{0,8,3,1,15,16,4,11,14,5,2,13,7,6,10,12,17,9}}

n=10:

{{-1,4,3,-5,-7,8,6,9,-2,10,2,-9,-6,-8,7,5,-3,-4,1},

{0,19,3,6,1,14,2,8,17,15,5,7,18,12,4,11,16,13,9,10}}

{{-1,-6,8,-5,-4,2,3,-9,7,10,-7,9,-3,-2,4,5,-8,6,1},

{0,19,13,1,16,12,14,17,8,15,5,18,7,4,2,6,11,3,9,10}}

{{2,4,-5,-7,3,-9,1,-6,-8,10,8,6,-1,9,-3,7,5,-4,-2},

{0,2,6,1,14,17,8,9,3,15,5,13,19,18,7,4,11,16,12,10}}

{{-2,-4,9,-1,7,-3,-5,6,8,10,-8,-6,5,3,-7,1,-9,4,2},

{0,18,14,3,2,9,6,1,7,15,5,17,11,16,19,12,13,4,8,10}}

{{2,5,-9,-4,-1,-7,3,-8,-6,10,6,8,-3,7,1,4,9,-5,-2},

{0,2,7,18,14,13,6,9,1,15,5,11,19,16,3,4,8,17,12,10}}

{{2,7,-3,5,6,-9,-4,-1,-8,10,8,1,4,9,-6,-5,3,-7,-2},

{0,2,9,6,11,17,8,4,3,15,5,13,14,18,7,1,16,19,12,10}}

{{-2,-7,5,1,-3,8,-9,-4,6,10,-6,4,9,-8,3,-1,-5,7,2},

{0,18,11,16,17,14,2,13,9,15,5,19,3,12,4,7,6,1,8,10}}

{{-2,-7,5,-9,-3,8,1,6,-4,10,4,-6,-1,-8,3,9,-5,7,2},

{0,18,11,16,7,4,12,13,19,15,5,9,3,2,14,17,6,1,8,10}}

{{-2,8,3,5,-1,4,-6,-9,-7,10,7,9,6,-4,1,-5,-3,-8,2},

{0,18,6,9,14,13,17,11,2,15,5,12,1,7,3,4,19,16,8,10}}

{{2,-8,-3,6,-1,7,5,-9,-4,10,4,9,-5,-7,1,-6,3,8,-2},

{0,2,14,11,17,16,3,8,19,15,5,9,18,13,6,7,1,4,12,10}}

{{2,-8,7,-4,-1,-3,5,-9,6,10,-6,9,-5,3,1,4,-7,8,-2},

{0,2,14,1,17,16,13,18,9,15,5,19,8,3,6,7,11,4,12,10}}

{{-2,-9,-6,-1,4,5,-7,3,8,10,-8,-3,7,-5,-4,1,6,9,2},

{0,18,9,3,2,6,11,4,7,15,5,17,14,1,16,12,13,19,8,10}}

{{2,-9,-7,-8,1,-5,3,-6,4,10,-4,6,-3,5,-1,8,7,9,-2},

{0,2,13,6,18,19,14,17,11,15,5,1,7,4,9,8,16,3,12,10}}

{{-3,-1,-4,6,-9,5,7,2,-8,10,8,-2,-7,-5,9,-6,4,1,3},

{0,17,16,12,18,9,14,1,3,15,5,13,11,4,19,8,2,6,7,10}}

{{3,-1,7,8,-6,-5,2,-4,-9,10,9,4,-2,5,6,-8,-7,1,-3},

{0,3,2,9,17,11,6,8,4,15,5,14,18,16,1,7,19,12,13,10}}

{{-3,-5,-1,-8,-7,-2,4,-9,6,10,-6,9,-4,2,7,8,1,5,3},

{0,17,12,11,3,16,14,18,9,15,5,19,8,4,6,13,1,2,7,10}}

{{-3,-8,5,4,-7,2,9,-6,-1,10,1,6,-9,-2,7,-4,-5,8,3},

{0,17,9,14,18,11,13,2,16,15,5,6,12,3,1,8,4,19,7,10}}

{{3,8,5,6,7,-2,-9,-4,1,10,-1,4,9,2,-7,-6,-5,-8,-3},

{0,3,11,16,2,9,7,18,14,15,5,4,8,17,19,12,6,1,13,10}}

{{-3,-9,-7,-8,-4,-5,-2,-6,-1,10,1,6,2,5,4,8,7,9,3},

{0,17,8,1,13,9,4,2,16,15,5,6,12,14,19,3,11,18,7,10}}

{{4,-1,5,-7,-9,-6,3,-2,8,10,-8,2,-3,6,9,7,-5,1,-4},

{0,4,3,8,1,12,6,9,7,15,5,17,19,16,2,11,18,13,14,10}}

{{-4,1,5,7,9,6,-3,-8,2,10,-2,8,3,-6,-9,-7,-5,-1,4},

{0,16,17,2,9,18,4,1,13,15,5,3,11,14,8,19,12,7,6,10}}

{{4,-1,6,-8,-3,-2,-9,-5,-7,10,7,5,9,2,3,8,-6,1,-4},

{0,4,3,9,1,18,16,7,2,15,5,12,17,6,8,11,19,13,14,10}}

{{-4,-2,7,-3,-9,-6,-1,5,8,10,-8,-5,1,6,9,3,-7,2,4},

{0,16,14,1,18,9,3,2,7,15,5,17,12,13,19,8,11,4,6,10}}

{{4,-6,-1,2,-7,9,5,-3,-8,10,8,3,-5,-9,7,-2,1,6,-4},

{0,4,18,17,19,12,1,6,3,15,5,13,16,11,2,9,7,8,14,10}}

{{-4,6,1,8,7,-9,5,3,-2,10,2,-3,-5,9,-7,-8,-1,-6,4},

{0,16,2,3,11,18,9,14,17,15,5,7,4,19,8,1,13,12,6,10}}

{{4,-6,3,-5,1,-8,-7,-9,2,10,-2,9,7,8,-1,5,-3,6,-4},

{0,4,18,1,16,17,9,2,13,15,5,3,12,19,7,6,11,8,14,10}}

{{-4,-7,-8,-3,6,-1,9,5,-2,10,2,-5,-9,1,-6,3,8,7,4},

{0,16,9,1,18,4,3,12,17,15,5,7,2,13,14,8,11,19,6,10}}

{{4,9,5,-7,1,-6,3,8,-2,10,2,-8,-3,6,-1,7,-5,-9,-4},

{0,4,13,18,11,12,6,9,17,15,5,7,19,16,2,1,8,3,14,10}}

{{-4,-9,-8,-5,7,-3,-6,1,2,10,-2,-1,6,3,-7,5,8,9,4},

{0,16,7,19,14,1,18,12,13,15,5,3,2,8,11,4,9,17,6,10}}

{{-6,-1,5,3,-9,4,-7,-2,8,10,-8,2,7,-4,9,-3,-5,1,6},

{0,14,13,18,1,12,16,9,7,15,5,17,19,6,2,11,8,3,4,10}}

{{-6,4,1,-8,2,-7,-9,-5,3,10,-3,5,9,7,-2,8,-1,-4,6},

{0,14,18,19,11,13,6,17,12,15,5,2,7,16,3,1,9,8,4,10}}

{{-6,4,3,-5,-9,-8,-7,1,2,10,-2,-1,7,8,9,5,-3,-4,6},

{0,14,18,1,16,7,19,12,13,15,5,3,2,9,17,6,11,8,4,10}}

{{6,-4,7,-5,9,-2,-3,-1,8,10,-8,1,3,2,-9,5,-7,4,-6},

{0,6,2,9,4,13,11,8,7,15,5,17,18,1,3,14,19,12,16,10}}

{{-6,4,9,2,-7,-1,5,-3,-8,10,8,3,-5,1,7,-2,-9,-4,6},

{0,14,18,7,9,2,1,6,3,15,5,13,16,11,12,19,17,8,4,10}}

{{6,-7,5,8,9,-3,-1,-4,2,10,-2,4,1,3,-9,-8,-5,7,-6},

{0,6,19,4,12,1,18,17,13,15,5,3,7,8,11,2,14,9,16,10}}

{{-6,9,5,3,1,4,-7,8,-2,10,2,-8,7,-4,-1,-3,-5,-9,6},

{0,14,3,8,11,12,16,9,17,15,5,7,19,6,2,1,18,13,4,10}}

{{6,-9,5,7,-1,-4,-3,-8,2,10,-2,8,3,4,1,-7,-5,9,-6},

{0,6,17,2,9,8,4,1,13,15,5,3,11,14,18,19,12,7,16,10}}

{{-7,-2,-4,-5,-8,-6,-9,-3,-1,10,1,3,9,6,8,5,4,2,7},

{0,13,11,7,2,14,8,19,16,15,5,6,9,18,4,12,17,1,3,10}}

{{7,4,1,6,8,3,-5,9,2,10,-2,-9,5,-3,-8,-6,-1,-4,-7},

{0,7,11,12,18,6,9,4,13,15,5,3,14,19,16,8,2,1,17,10}}

{{7,-5,-1,-3,8,-2,9,6,-4,10,4,-6,-9,2,-8,3,1,5,-7},

{0,7,2,1,18,6,4,13,19,15,5,9,3,14,16,8,11,12,17,10}}

{{-7,8,3,-6,-9,-2,5,4,-1,10,1,-4,-5,2,9,6,-3,-8,7},

{0,13,1,4,18,9,7,12,16,15,5,6,2,17,19,8,14,11,3,10}}

{{8,-1,-3,-2,9,-5,7,-4,6,10,-6,4,-7,5,-9,2,3,1,-8},

{0,8,7,4,2,11,6,13,9,15,5,19,3,16,1,12,14,17,18,10}}

{{-8,1,4,2,-3,-5,7,6,-9,10,9,-6,-7,5,3,-2,-4,-1,8},

{0,12,13,17,19,16,11,18,4,15,5,14,8,1,6,9,7,3,2,10}}

{{8,1,-5,7,2,4,9,6,3,10,-3,-6,-9,-4,-2,-7,5,-1,-8},

{0,8,9,4,11,13,17,6,12,15,5,2,16,7,3,1,14,19,18,10}}

{{-8,2,-3,6,9,7,5,1,-4,10,4,-1,-5,-7,-9,-6,3,-2,8},

{0,12,14,11,17,6,13,18,19,15,5,9,8,3,16,7,1,4,2,10}}

{{8,-2,-7,4,1,3,5,9,-6,10,6,-9,-5,-3,-1,-4,7,2,-8},

{0,8,6,19,3,4,7,12,1,15,5,11,2,17,14,13,9,16,18,10}}

{{-8,2,7,-4,9,-3,5,1,6,10,-6,-1,-5,3,-9,4,-7,-2,8},

{0,12,14,1,17,6,3,8,9,15,5,19,18,13,16,7,11,4,2,10}}

{{8,-2,-7,-6,-9,3,5,-1,4,10,-4,1,-5,-3,9,6,7,2,-8},

{0,8,6,19,13,4,7,12,11,15,5,1,2,17,14,3,9,16,18,10}}

{{-8,5,1,-9,4,-7,-2,-3,-6,10,6,3,2,7,-4,9,-1,-5,8},

{0,12,17,18,9,13,6,4,1,15,5,11,14,16,3,19,8,7,2,10}}

{{-8,-6,-5,-7,3,-9,1,4,2,10,-2,-4,-1,9,-3,7,5,6,8},

{0,12,6,1,14,17,8,9,13,15,5,3,19,18,7,4,11,16,2,10}}

{{8,6,9,-1,7,-3,-5,-4,-2,10,2,4,5,3,-7,1,-9,-6,-8},

{0,8,14,3,2,9,6,1,17,15,5,7,11,16,19,12,13,4,18,10}}

{{8,-6,-9,-7,1,2,5,-3,4,10,-4,3,-5,-2,-1,7,9,6,-8},

{0,8,2,13,6,7,9,14,11,15,5,1,4,19,17,16,3,12,18,10}}

{{-8,7,-3,5,-4,-9,6,-1,2,10,-2,1,-6,9,4,-5,3,-7,8},

{0,12,19,16,1,17,8,14,13,15,5,3,4,18,7,11,6,9,2,10}}

{{-8,7,5,-1,3,2,9,4,-6,10,6,-4,-9,-2,-3,1,-5,-7,8},

{0,12,19,4,3,6,8,17,1,15,5,11,7,18,16,13,14,9,2,10}}

{{8,9,-3,-2,-1,-5,7,6,-4,10,4,-6,-7,5,1,2,3,-9,-8},

{0,8,17,14,12,11,6,13,19,15,5,9,3,16,1,2,4,7,18,10}}

{{8,-9,4,-1,-6,5,-7,3,-2,10,2,-3,7,-5,6,1,-4,9,-8},

{0,8,19,3,2,16,1,14,17,15,5,7,4,11,6,12,13,9,18,10}}

{{-8,-9,4,2,7,-5,-3,6,1,10,-1,-6,3,5,-7,-2,-4,9,8},

{0,12,3,7,9,16,11,8,14,15,5,4,18,1,6,19,17,13,2,10}}

{{8,9,-4,-7,3,-5,-2,-1,-6,10,6,1,2,5,-3,7,4,-9,-8},

{0,8,17,13,6,9,4,2,1,15,5,11,12,14,19,16,3,7,18,10}}

{{-9,-4,1,8,-3,6,5,-2,-7,10,7,2,-5,-6,3,-8,-1,4,9},

{0,11,7,8,16,13,19,4,2,15,5,12,14,9,3,6,18,17,1,10}}

{{9,4,-7,-5,3,8,6,-1,-2,10,2,1,-6,-8,-3,5,7,-4,-9},

{0,9,13,6,1,4,12,18,17,15,5,7,8,2,14,11,16,3,19,10}}

{{-9,-4,-8,-5,-6,-2,-3,-1,-7,10,7,1,3,2,6,5,8,4,9},

{0,11,7,19,14,8,6,3,2,15,5,12,13,16,18,4,9,17,1,10}}

{{9,-6,-1,-8,3,4,5,2,7,10,-7,-2,-5,-4,-3,8,1,6,-9},

{0,9,3,2,14,17,1,6,8,15,5,18,16,11,7,4,12,13,19,10}}

{{-9,6,5,-8,-1,-4,7,2,-3,10,3,-2,-7,4,1,8,-5,-6,9},

{0,11,17,2,14,13,9,16,18,15,5,8,6,19,3,4,12,7,1,10}}

{{-9,-7,-1,-4,-2,-5,-6,-8,-3,10,3,8,6,5,2,4,1,7,9},

{0,11,4,3,19,17,12,6,18,15,5,8,16,2,7,9,13,14,1,10}}

n=11: (partial list)

{{-1,-3,-6,-10,2,4,-5,-9,-7,8,11,-8,7,9,5,-4,-2,10,6,3,1},

{0,21,18,12,2,4,8,3,16,9,17,6,20,5,14,19,15,13,1,7,10,11}}

{{-1,-3,-10,-4,-6,-7,-8,-2,9,5,11,-5,-9,2,8,7,6,4,10,3,1},

{0,21,18,8,4,20,13,5,3,12,17,6,1,14,16,2,9,15,19,7,10,11}}

{{-1,4,-9,-3,7,-5,8,6,-10,-2,11,2,10,-6,-8,5,-7,3,9,-4,1},

{0,21,3,16,13,20,15,1,7,19,17,6,8,18,12,4,9,2,5,14,10,11}}

{{-1,5,3,-9,-6,10,-8,-4,7,-2,11,2,-7,4,8,-10,6,9,-3,-5,1},

{0,21,4,7,20,14,2,16,12,19,17,6,8,1,5,13,3,9,18,15,10,11}}

n=12: (one example)

{{3,-2,-6,-8,11,-5,9,7,-1,-4,-10,12,10,4,1,-7,-9,5,-11,8,6,2,-3},

{0,3,1,19,11,22,17,2,9,8,4,18,6,16,20,21,14,5,10,23,7,13,15,12}}

n=13: (one example)

{3,1,8,7,9,6,2,-12,-10,-5,-4,-11,13,11,4,5,10,12,-2,-6,-9,-7,-8,-1,-3},

{0,3,4,12,19,2,8,10,24,14,9,5,20,7,18,22,1,11,23,21,15,6,25,17,16,13}}