Triple-free subsets (A050296)

[John Layman]

I have recently used a rather more efficient algorithm to evaluate Eric Weisstein's 
A050296 in the OEIS:

ID Number: A050296
Sequence:  1,1,2,2,3,4,5,5,6,6,7,7,8,8,9,10,11,11,12,12
Name:      Maximum cardinality of a strongly triple-free subset of {1, 2, ..., n}.
Links:     E. W. Weisstein, Link to a section of The World of Mathematics.
See also:  Cf. A050291-A050295.
Keywords:  nonn,more
Offset:    1
Author(s): Eric W. Weisstein (eric at weisstein.com)


Thus I am able to calculate up to 500-1000 terms without an unreasonable computation
time and have submitted an extension to Neil.

The reason for this note is that my calculated values for a(n)/n do not seem to
be tending toward S. Finch's limiting value 0.613475... as stated in Eric's link 
to "The World of Mathematics".  (Finch's paper is currently unavailable on the web.)

I would appreciate it if someone would confirm my calculated values below.

For a(1) - a(700) I get:

   1   1   2   2   3   4   5   5   6   6   7   7   8   8   9  10  11  11  12  12
  13  13  14  15  16  16  16  16  17  18  19  19  19  19  20  21  22  22  22  22
  23  24  25  26  27  27  28  28  29  29  29  30  31  32  33  33  33  33  34  34
  35  35  36  37  38  39  40  41  41  41  42  42  43  43  43  44  45  46  47  48
  49  49  50  50  51  51  52  52  53  53  54  55  55  55  56  57  58  58  59  60
  61  62  63  63  63  63  64  64  65  65  65  66  67  68  69  70  71  71  72  73
  74  74  74  75  76  76  77  77  77  77  78  78  79  79  79  79  80  81  82  83
  83  83  84  85  86  86  86  87  88  89  90  90  91  91  92  92  93  93  93  93
  94  94  95  96  96  96  97  98  99  99 100 100 101 102 103 104 104 104 105 106
 107 107 107 107 108 109 110 111 111 111 112 112 113 113 113 114 115 115 116 116
 116 116 117 117 118 118 119 120 121 122 123 124 124 124 125 126 127 127 127 128
 129 130 131 131 132 132 133 133 134 134 134 134 135 135 136 137 137 137 138 138
 139 139 139 140 141 142 143 143 143 143 144 145 146 146 146 147 148 149 150 151
 152 152 153 154 155 155 155 156 157 158 159 160 160 160 161 161 162 162 163 163
 164 165 166 167 167 167 168 168 169 169 169 170 171 172 173 173 173 173 174 174
 175 175 175 176 177 177 178 179 179 179 180 181 182 182 183 184 185 186 187 188
 188 188 189 190 191 191 191 191 192 193 194 195 196 196 197 197 198 198 198 199
 200 200 201 201 201 201 202 202 203 203 203 203 204 205 206 207 207 207 208 208
 209 209 209 210 211 212 213 214 215 215 216 216 217 217 217 217 218 219 220 221
 221 221 222 223 224 224 225 226 227 228 229 229 229 229 230 231 232 232 232 233
 234 235 236 237 238 238 239 240 241 241 241 242 243 243 244 244 244 244 245 245
 246 246 247 247 248 249 250 251 251 251 252 252 253 253 253 254 255 256 257 257
 258 258 259 259 260 260 260 261 262 262 263 264 264 264 265 266 267 267 267 268
 269 270 271 272 272 272 273 274 275 275 275 275 276 277 278 279 280 280 281 282
 283 283 283 284 285 286 287 287 287 287 288 288 289 289 290 291 292 293 294 295
 295 295 296 296 297 297 297 298 299 300 301 301 301 301 302 302 303 303 304 304
 305 305 306 307 307 307 308 308 309 309 310 311 312 313 314 314 314 314 315 315
 316 316 316 316 317 318 319 320 321 321 322 323 324 324 324 325 326 326 327 328
 328 328 329 329 330 330 331 331 332 333 334 335 335 335 336 337 338 338 338 339
 340 341 342 342 343 343 344 344 345 345 345 346 347 348 349 350 350 350 351 352
 353 353 354 355 356 357 358 358 358 358 359 360 361 361 361 361 362 363 364 365
 365 365 366 366 367 367 367 368 369 369 370 370 370 370 371 371 372 372 373 373
 374 375 376 377 377 377 378 378 379 379 379 380 381 382 383 384 385 385 386 386
 387 387 387 387 388 388 389 390 390 390 391 392 393 393 393 394 395 396 397 397
 397 397 398 399 400 400 400 401 402 403 404 405 406 406 407 408 409 409 409 410


For the ratios a(10k)/(10k) I get (for k=1..70):

0.60000   0.60000   0.60000   0.55000   0.58000   0.56667   0.58571   0.60000   
0.58889   0.60000   0.59091   0.60833   0.59231   0.59286   0.59333   0.58125   
0.58235   0.58889   0.58421   0.58000   0.58095   0.58182   0.58261   0.57500   
0.57200   0.58077   0.58519   0.58214   0.58276   0.58000   0.57742   0.58750   
0.58485   0.58529   0.58000   0.57778   0.58108   0.58158   0.58462   0.58250   
0.58780   0.58333   0.58372   0.58409   0.58222   0.58261   0.58511   0.58750   
0.58571   0.59000   0.58824   0.58462   0.58302   0.58333   0.58364   0.58571   
0.58421   0.58448   0.58475   0.58667   0.58689   0.58871   0.58571   0.58281   
0.58308   0.58485   0.58209   0.58382   0.58406   0.58571


P.S.  For the analogous weakly triple-free subsets in A050294, I find a strong 
tendency toward a limiting value of 0.75 for a(n)/n, instead of Finch's
asymptotic value 0.8.  If my calculated values are confirmed, then it would
seem that the theory needs to be revisited.

John