You can use the Medjig method of Willem Barink (see the 6x6 magic square) to construct a 16x16 magic square. The first grid consists of 'the 2x2 blown up' version of the 8x8 magic square and the second grid consist of the 2x2 Medjig tiles. If you use a Franklin panmagic 8x8 square and a tight grid of Medjig tiles you can construct a panmagic 16x16 square.
Take a number from a cell of the first grid and add 64 x number from the same cell of the second grid.
+ 1x number
1 | 1 | 56 | 56 | 27 | 27 | 46 | 46 | 17 | 17 | 40 | 40 | 11 | 11 | 62 | 62 |
1 | 1 | 56 | 56 | 27 | 27 | 46 | 46 | 17 | 17 | 40 | 40 | 11 | 11 | 62 | 62 |
63 | 63 | 10 | 10 | 37 | 37 | 20 | 20 | 47 | 47 | 26 | 26 | 53 | 53 | 4 | 4 |
63 | 63 | 10 | 10 | 37 | 37 | 20 | 20 | 47 | 47 | 26 | 26 | 53 | 53 | 4 | 4 |
38 | 38 | 19 | 19 | 64 | 64 | 9 | 9 | 54 | 54 | 3 | 3 | 48 | 48 | 25 | 25 |
38 | 38 | 19 | 19 | 64 | 64 | 9 | 9 | 54 | 54 | 3 | 3 | 48 | 48 | 25 | 25 |
28 | 28 | 45 | 45 | 2 | 2 | 55 | 55 | 12 | 12 | 61 | 61 | 18 | 18 | 39 | 39 |
28 | 28 | 45 | 45 | 2 | 2 | 55 | 55 | 12 | 12 | 61 | 61 | 18 | 18 | 39 | 39 |
33 | 33 | 24 | 24 | 59 | 59 | 14 | 14 | 49 | 49 | 8 | 8 | 43 | 43 | 30 | 30 |
33 | 33 | 24 | 24 | 59 | 59 | 14 | 14 | 49 | 49 | 8 | 8 | 43 | 43 | 30 | 30 |
31 | 31 | 42 | 42 | 5 | 5 | 52 | 52 | 15 | 15 | 58 | 58 | 21 | 21 | 36 | 36 |
31 | 31 | 42 | 42 | 5 | 5 | 52 | 52 | 15 | 15 | 58 | 58 | 21 | 21 | 36 | 36 |
6 | 6 | 51 | 51 | 32 | 32 | 41 | 41 | 22 | 22 | 35 | 35 | 16 | 16 | 57 | 57 |
6 | 6 | 51 | 51 | 32 | 32 | 41 | 41 | 22 | 22 | 35 | 35 | 16 | 16 | 57 | 57 |
60 | 60 | 13 | 13 | 34 | 34 | 23 | 23 | 44 | 44 | 29 | 29 | 50 | 50 | 7 | 7 |
60 | 60 | 13 | 13 | 34 | 34 | 23 | 23 | 44 | 44 | 29 | 29 | 50 | 50 | 7 | 7 |
+ 64 x number
0 | 3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 | 3 |
1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 |
3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 |
2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 |
0 | 3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 | 3 |
1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 |
3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 |
2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 |
0 | 3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 | 3 |
1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 |
3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 |
2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 |
0 | 3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 | 3 |
1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 |
3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 |
2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 |
= 16x16 panmagic square
1 | 193 | 56 | 248 | 27 | 219 | 46 | 238 | 17 | 209 | 40 | 232 | 11 | 203 | 62 | 254 |
65 | 129 | 120 | 184 | 91 | 155 | 110 | 174 | 81 | 145 | 104 | 168 | 75 | 139 | 126 | 190 |
255 | 63 | 202 | 10 | 229 | 37 | 212 | 20 | 239 | 47 | 218 | 26 | 245 | 53 | 196 | 4 |
191 | 127 | 138 | 74 | 165 | 101 | 148 | 84 | 175 | 111 | 154 | 90 | 181 | 117 | 132 | 68 |
38 | 230 | 19 | 211 | 64 | 256 | 9 | 201 | 54 | 246 | 3 | 195 | 48 | 240 | 25 | 217 |
102 | 166 | 83 | 147 | 128 | 192 | 73 | 137 | 118 | 182 | 67 | 131 | 112 | 176 | 89 | 153 |
220 | 28 | 237 | 45 | 194 | 2 | 247 | 55 | 204 | 12 | 253 | 61 | 210 | 18 | 231 | 39 |
156 | 92 | 173 | 109 | 130 | 66 | 183 | 119 | 140 | 76 | 189 | 125 | 146 | 82 | 167 | 103 |
33 | 225 | 24 | 216 | 59 | 251 | 14 | 206 | 49 | 241 | 8 | 200 | 43 | 235 | 30 | 222 |
97 | 161 | 88 | 152 | 123 | 187 | 78 | 142 | 113 | 177 | 72 | 136 | 107 | 171 | 94 | 158 |
223 | 31 | 234 | 42 | 197 | 5 | 244 | 52 | 207 | 15 | 250 | 58 | 213 | 21 | 228 | 36 |
159 | 95 | 170 | 106 | 133 | 69 | 180 | 116 | 143 | 79 | 186 | 122 | 149 | 85 | 164 | 100 |
6 | 198 | 51 | 243 | 32 | 224 | 41 | 233 | 22 | 214 | 35 | 227 | 16 | 208 | 57 | 249 |
70 | 134 | 115 | 179 | 96 | 160 | 105 | 169 | 86 | 150 | 99 | 163 | 80 | 144 | 121 | 185 |
252 | 60 | 205 | 13 | 226 | 34 | 215 | 23 | 236 | 44 | 221 | 29 | 242 | 50 | 199 | 7 |
188 | 124 | 141 | 77 | 162 | 98 | 151 | 87 | 172 | 108 | 157 | 93 | 178 | 114 | 135 | 71 |
N.B.: Each 1/2 row/colum/diagonal gives 1/2 of the magic sum (1/2 x 2056 = 1028) and each random chosen 4x4 subsquare inside the 16x16 magic square gives the magic sum of 2056.