[M2] (golly 2.0) #C Patterns whose population grows logarithmically have #C long been known, but this is believed to be the first pattern #C whose *diameter* grows logarithmically. It is based on a #C blinker puffer that can be deleted by its own output (a blinker), #C or push a block out by 8 cells. #C #C By Jason Summers with Gabriel Nivasch, 16 Aug 03 #C #C Analysis by Dean Hickerson: #C #C Let g(n) be the generation in which the leading puffer is in the #C same position as in gen 0, the block is 8n units to the right of #C its initial position, and there are no blinkers in between. I.e., #C the puffer is on its way to push the block from position n to #C position n+1. g(n) is always a multiple of 240; the first few #C values are: #C n 0 1 2 3 4 5 6 7 8 9 10 #C g(n) 0 240 720 1440 2400 3840 6000 9120 13680 20400 30240 #C g(n)/240 0 1 3 6 10 16 25 38 57 85 126 #C #C Letting h(n)=g(n)/240, we have the recurrence #C h(n) = h(n-1) + h(n-3) + 3. #C #C Also, #C h(n) = round(C * R^n) - 3, #C #C where #C R = (2 + cbrt(116 + 12 sqrt(93)) + cbrt(116 - 12 sqrt(93))) / 6 #C ~ 1.465571231876768026656731225219939108025577568472285701643 #C #C and #C C = 1 + cbrt((13113 + 151 sqrt(93))/17298) + #C cbrt((13113 - 151 sqrt(93))/17298) #C ~ 2.82110012372929333835936935875070702572316547898891137253679 #C (C satisfies 31 C^3 - 93 C^2 + 16C - 1 = 0.) #C #C Since the diameter in generation g(n) is 8n + O(1), we find that #C the diameter in gen t is asymptotic to A * log(t), where #C A = 8/log(R) #C ~ 20.92898063664810844728999060141708775230924963625537466402 $$$$$$$....**$ 4 0 0 1 0 5 0 0 2 0 6 0 0 0 3 7 0 0 0 4 8 0 0 0 5 9 0 0 6 0 10 0 0 7 0 11 0 0 8 0 $$$$$$$.**$ 4 0 0 0 10 ....**$....**$$$$$$...***$ .**$ $...*.*$..*****$.**...**$.**...**$$$....*$ 4 12 13 14 0 5 0 11 0 15 $$$$.....*$.....*.*$......*$......*$ $$$$......*$......*$*.....*$.*$ 4 0 0 17 18 $$..**$.***...*$**.....*$.*....*$**$.***$ $$.*.....*$...*...*$$....*$...**..*$.......*$ 4 0 0 20 21 ......*$.....*.*$.....*$ *$$.......*$$.......*$ ......**$.....***$....****$...**$....****$.....***$......**$ **$***$****$...**$****$***$**$ 4 23 24 25 26 ..**$$$**$*.....*$......*$......*$.......*$ $$$.....**$*....*$.*....*$.*.....*$**$ $.....**$.....**$....*$....*.**$.....***$......*$ ***$***$.**$ 4 28 29 30 31 5 19 22 27 32 6 0 0 16 33 $$*$*$$*$.*$.*$ 4 0 0 35 0 *$$$*.**$...*$..*$**$ $$$$$$.......*$.......*$ $$$$.*$*.*$...**$...**$ 4 37 0 38 39 $$$$$$$**$ 4 0 0 41 0 5 36 0 40 42 6 0 0 43 0 $$$$....**$....**$ $$......*$....*.*$..**$..**$..**$....*.*$ $$$$$**$**$ ......*$$$.....*$....*.*$....**.*$....**.*$....**.*$ 4 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$..**$..*.*$..*$ 4 0 0 103 104 .**.**$.**.*$.*.*$..*$......**$......**$ .**$.**...**$......**$ ......**$......**$$$$....**$.....***$......**$ *$*$$$$.**$**$*$ 4 106 107 108 109 $.*$*.*$$$***$***$.*$ $$.*$***$***$$$*.*$ 4 0 111 0 112 .......*$.***$.*$..*$ $....***$$....*.*$...*****$..**...*$..**...*$ $$$$$*$*$ 4 114 0 115 116 5 105 110 113 117 6 89 102 0 118 7 34 44 84 119 $$$$$$$....*$ 4 0 0 0 121 ...***$..**.**$.***.***$.***.***$.***.***$.***.***$..**.**$...***$ ....*$ 4 0 123 0 124 5 122 0 125 0 6 0 0 0 126 $$.......*$......*$$.....*$.....*$ $***$...*$....*$$.....*$.....*$ 4 0 0 128 129 ......*$.......*$ ....*$...*$***$ $....*.*$....**$.....*$ 4 131 132 0 133 5 0 130 0 134 ....**$$$$.....*$....***$...*...*$..*.***$ $$$$$$$*$ ...*****$$$$$$....*..*$....*.*$ 4 136 137 138 51 ...**$..**$..**.*$...***$$$....**$.......*$ $$$$$$.**$ ....*$.....**$......*$.......*$.......*$ ..*$**$*$$.....**$.....**$ 4 140 141 142 143 $$.......*$......*$......*$......*$.......*$ 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243 0 ......*$....***$ 4 62 0 245 0 5 238 242 244 246 $$$*****$.....*$**...*$....*$ 4 0 0 248 0 5 249 0 0 0 6 234 0 247 250 4 0 0 0 172 5 0 0 0 252 $$$$$.......*$.....*$....*$ $$$$$*$*$....****$ 4 0 0 254 255 $$$$$....**$....**$*$ 4 0 0 257 0 5 0 0 256 258 $$......**$......**$ 4 0 260 0 0 4 0 187 0 171 5 261 262 0 0 ....*$....*$.....*$.......*$$...*.*$....**$....*$ ...*$....**$*$*$ $$$$...*$....**$...**$ 4 264 265 174 266 .*$.*...*$*...***$...*...*$.....*$..*$..*$...*...*$ ....***$ 4 268 116 269 62 $$$.......*$.......*$ 4 0 271 0 0 $$$*$*$ 4 273 180 0 243 5 267 270 272 274 6 253 259 263 275 $$$$$$**$.**$ $$.......*$......**$ 4 277 0 81 278 $$.**$*.*$$..*$.*$ $....*.*$..*...*$..*$.*....*$..*$..*...*$....*.*$ 4 0 0 280 281 .....*$.....*.*$......**$$$$.....**$.....*.*$ $.......*$....****$...****$...*..*$...****$....****$.......*$ 4 1 0 283 284 5 0 279 282 285 $$*$$*$ 4 0 0 287 0 $$$.***$ 4 0 0 0 289 $$$$$$$.......*$ $$$$$$$..**$ 4 0 0 291 292 5 288 0 290 293 $......**$......**$.....*$....*$.....*$......**$ $*$*$.*$..*$.*$*$ $$$$$..**$..**$ 4 295 296 297 0 $*$ 4 212 299 0 0 ......**$......**$ 4 301 0 0 0 5 298 300 302 0 $......**$......*$.......*$ ...*****$..*.***$...*...*$....***$.....*$ $*$$$$.**..**$.**..**$.....**$ 4 0 304 305 306 ......*$*......*$ ..***..*$..**$ ......**$......*$......*$..*$..**$...**$...***$...**$ $*$$......*$......*$......*$ 4 308 309 310 311 .....**$.....**$ 4 313 0 0 0 ..**$..*$ 4 315 0 0 0 5 307 312 314 316 6 286 294 303 317 4 0 90 0 0 $$$$$$$**..*$ 4 0 0 320 0 $$......**$....*$...*$..*$..*$..*$ 4 0 0 0 322 5 0 319 321 323 $$$.*$..*$...*$...*$...*$ $$$$$......**$.....*.*$.......*$ 4 0 0 325 326 $.***$ .......*$......**$.....***$......**$.......*$ 4 0 0 328 329 5 95 0 327 330 **...*$**..*$ $..**$.***$**$.*....*$**.....*$.***...*$..**$ $$$...**$....*$$...*$.*$ 4 332 0 333 334 ...*$....*$......**$$$$$.....*$ $**$**$ ....*.*$....**.*$....**.*$....**.*$....*.*$.....*$ 4 0 336 337 338 5 335 339 0 0 ..*$.*$$$$......**$$.......*$ $$*......*$......**$....**$......**$.......*$ $*.*$..*$$.*$.....**$..*....*$*.*$ 4 341 116 342 343 $$.......*$......**$.....***$....*.*$....**$ $$$*$**$*.*$.**$ $.......*$......*$..**..*$..**...*$......**$......**$......**$ $$*$*$ 4 345 346 347 348 5 344 349 0 0 6 324 331 340 350 7 251 276 318 351 $*$.*...**$*....**$ $$$$$$.*..*$.....*$ 4 353 0 354 0 5 0 0 355 0 6 0 0 356 0 $.......*$.......*$ ......**$*.....*$*.....**$.......*$ 4 0 291 358 359 5 0 0 0 360 $$$$$$..*$***...**$ $$$$......**$......*$.......*$ **...**$.*...**$**....*$***...*$..*$ .......*$*.**..*$*.**..**$**$ 4 362 363 364 365 $$$$$*$**$***$ $$$$.....**$.....**$$.**$ **$*$$$$$$*$ .**$$$$$$......*$.....***$ 4 367 368 369 370 5 0 0 366 371 $$$$...*$.**.****$...*$ $$$$*$.**$*$ 4 373 374 0 0 .**$**$ ....****$...**$....****$....*$.....*.*$......*$ $*$.*$**$ 4 376 377 0 378 4 0 0 243 354 5 375 379 0 380 6 361 372 0 381 .*...*$..****$$$..***$.*****$.***.**$....**$ 4 383 0 0 0 ***$.*.*...*$.*..*..*$.*.*$***$*$ $*$*$ 4 137 0 385 386 5 384 0 387 0 $$$$......**$$.....*$ 4 0 0 0 389 5 0 0 0 390 4 0 0 0 229 .....**$$$$$$$..**$ $.......*$*.....**$*......*$ ..**$*$*$*$..**$..**$ 4 0 393 394 395 $$$$$..**$.**$...*$ 4 0 0 397 292 5 392 396 122 398 6 388 391 0 399 $$$$$....*$...*.*$.**...*$ ....**$....**$$*$.**$$...*$ .**...*$.**...*$...*.*$....*$ 4 47 401 402 403 $$$...*$.**$..**$ ....*.*$...***$..***$...***$....*.*$.....**$ 4 0 59 405 406 5 0 197 404 407 4 0 383 0 0 $..**$.*..****$..**.***$ $$$$.....*$.....***$.....*$ 4 0 0 410 411 $$$$*.**.*$*.**.***$*.**.*$ $$$$*$*$*$ 4 0 0 413 414 5 198 409 412 415 *.**$$$$$.**$ $$$..*.*$..**$...*$ $$$.....**$.....**$ 4 417 418 419 0 $$$$$...*..*$...****$...*..*$ $$$$$**.*..*$**.****$**.*..*$ 4 421 422 0 0 5 420 423 0 0 $$$$$$.......*$......*$ ....**$....**$ .....***$....***$.....***$......*$.......*$ 4 0 425 426 427 $$....**$...*.*$.....*$ ....**$...*..**$....***$*....**$*......*$ ..*$.*.*$*...*$*...*$*...*$.*.*$..*$ 4 182 429 430 431 5 428 432 0 0 6 408 416 424 433 7 357 382 400 434 $...**..*$...**..*$.......*$ ....**$*....**$*....***$*....**$....**$....*$ $$$$$....**$....**$ 4 436 437 0 438 $$$$......**$......*$ ..*.*$..*$$$$*$*$**$ $$$$.*$..**$..**$..**$ 4 440 441 271 442 .*$$.....*$....*.*$...*...*$...*****$..**...*$...*****$ ....***$.....*$ 4 212 444 0 445 5 439 443 0 446 $$.....**$.....**$ 4 0 448 0 0 5 0 449 0 0 6 0 447 0 450 $$$$..*$..*$..*$ 4 0 0 452 0 $$$$$$..**$.**$ ...*$$$$$....**$...*.*$..***$ 4 51 454 10 455 $...*$.**.****$...*$ $*$.**$*$ $$$$..**.*$**.*.*$**.*.**$.....**$ $$$.*$.****$..****$..*..*$..****$ 4 457 458 459 460 5 453 0 456 461 $$$$$...**$...**$ 4 0 0 463 0 5 0 0 464 0 .***$..***$...*.*$....**$ 4 13 466 0 0 ....**$ .****$.*$ 4 468 469 0 0 5 467 470 0 0 6 462 465 471 0 7 451 472 0 0 8 352 435 0 473 9 226 474 0 0 $...***$...***$..*...*$.*.....*$..*...*$...***$ 4 0 45 0 476 5 0 0 0 477 $$$$$$....**$**.*..*$ $$$$.**$.*.*$....*$.*..*$ **..**$ ....*$.*.*$.**$......*$....**$.....**$ 4 479 480 481 482 $$$$$$*.*$..*....*$ ......**$....**$......**$.......*$ .....**$.*$$..*$*.*$ 4 291 484 485 486 5 0 0 483 487 *$**$*$*$$$....*$..*.*$ $$$$......**$......*$....*.*$....**$ **$**$**$..*.*$....*$ $$$.....**$....*.*$....*$...**$ 4 489 490 491 492 $$$$$$....*.*$..*...*$ ..*$.*....*$..*$..*...*$....*.*$ 4 494 0 495 0 $$$$$*$.*$**$ 4 0 0 291 497 $......*$.....*$.....***$ 4 0 0 499 0 5 493 496 498 500 $$$.....*.*$.....**$......*$ $$$$$.....**$ $.*$.*$.*$ 4 502 503 504 0 $$$$.*.*..*$*..*..*$.*.*..*$ $$$$*$.*.**$*$ 4 506 507 0 0 5 505 508 0 0 6 478 488 501 509 $$$$$$......**$.....***$ 4 0 0 0 511 $$$$..*$..**$...**$...***$ 4 0 0 227 513 $...**$...**$ ..*.**$..*..*$..*.**$.....***$......**$ $..**$..**$ $$$$...*..**$..*...**$...*..**$ 4 515 516 517 518 ..****$..*....*$...*...*$...**.*$ ...**$..**$..*$$$$$...*.*$ $$$$....**$*..***$....**$ ....**$....*$$$*$.*$*$ 4 520 521 522 523 5 512 514 519 524 $$$$.**$.**$ 4 0 0 59 526 .....**$ $.......*$$$$$...*$....**$ 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......**$....*$...*$..**.*$...*$....*$......**$ 4 0 54 260 668 5 0 669 0 0 ..*....*$**$ ..*$**$ $*.....*$.*$.*$.*$*$ $.***$...*...*$..*....*$$$$.......*$ 4 671 672 673 674 *$.**$*.*$..*$ ....*.*$.....*$$$..**...*$..**...*$...*****$....*.*$ $$$.*$..**$..**$..**$.*$ $....***$$$**$**$$....**$ 4 676 677 678 679 4 0 54 0 0 4 0 468 0 0 5 675 680 681 682 6 661 667 670 683 $$$$$$..**$...*$ ..**$..*.*$.....*$..*..*$.....*$..*.*$..**$ ...*.*$....**$$$.......*$...**$...*.*$.....*$ 4 0 685 686 687 ...*$...**$....**$....***$*...**$...**$...*$ 4 0 0 689 0 5 197 198 688 690 4 565 0 0 0 $$$$$.......*$**....*$**....*$ 4 0 0 0 693 $$$$$*......*$.*....*$.*....*$ $$$$$*$.*$.*$ 4 0 0 695 696 5 409 692 694 697 4 631 528 0 0 $$...*$..***$..***$$**...**$**...**$ $$...*$..*.*$.**....*$.**....*$.***$..*.*$ 4 0 700 0 701 ...**$ 4 0 703 0 0 5 699 702 0 704 ......*$.......*$$$$..**$..*.*$..*$ $$.*.*$.*...*$.....*$.*....*$.....*$.*...*$ 4 0 706 631 707 .*....*$*......*$ 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