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Different displays of a long low-discrepancy sequence

It would be good to have some long sequences with low discrepancy displayed nicely here (or on subsidiary pages) in tabular form, together with discussions about the structures that can be found in these sequences. For now, here is the current record not displayed in a particularly helpful way. It is a sequence of discrepancy 2 and length 1124.

The raw sequence

+ - + + - - - - + + - + + + - - + - + + - - - + - + + - + - - + + - + - - + - - + + - + + - - - + + + - - + - + - - + + + + - - + - - + + - - + + + - - + + - + - - + - - + + - + - - + - + + + - - + - + - - + + - - - + + - + + + - - - + - - + - + + + + - - - - + + - + + + - - + - - + - - + - + - + + - + + + - - - - + + + + - - - + + - + - - + - + + - - + - + + - + - + - + + - - - + + - + + - - + - - + - - + + + + - + - - + - - + + - - + - + + - - - + - + + - + + + - - + + - - + - - + - + + - - + - + + - + + - - + - - + - + + - - + + - + - + - - + - + + + - - + - + + - - - - + + - - + - + + - + + - - + - + + - - + + - + - - + - + - - + + - - + - + + + - + - - + - + - - + + + + - - - + + - + - - + - + + - - + - + + - + + - - + - - + - + + - - + + - - - + + - - + - + + - + + - - + - - + + + + - - - - + + - + + - + + - - + - - + + - - - + + - + + - - + - + + - + - - - + - + + - + + - - + - - + - + + - - + + - + + + - - - + - + + - + + - - + - - - + + + - - + - + - - + + + - + - + + - + - - - + + - + - + + - - + - - + - + + - - + - + - + + - + - + - - + - + + - - + - - + - + + + - - - + + - + + + - + - - + - + + - - + - + + - + - - - + - + + - + + - - + - - - + + + - - + - + + - - + + - + - + + - + - - - + - - + - + + - - + - + + - + + - - + - + + - + + - - + - - + - + + - - + - - + - + + - + - - + + - + + - - + - - + - + + - - + - + + - + + - - + - + + - - + + - + - - + - + + - - + - - + - + + - - + - + + - + + - + + - - + - + + - - + - - + - + + - - + - + - - + + + - + - - + - + + - - + - + - - + + + - - - + + - + + - - + + - + - + + - - + - - + - - + - + + - + - - + + + - + - - + - + + - - + - + - - + + - - + - + + - + + - - + + - + - - + - - + - + + - + + - - + - + + - + + - - + - - + + + - - - + - - + - + + - - + + + + - + - - + + - - + - + + - - + - - + - + + - - + - + + - + + - - + - - + - + + - - - + + - - + + - - + - + + - + + - - + - - + - + + + - + - - + - + + - - + - + + - + - + - - + - + - + + - - + - + - - + + + - + - - - + + + - - + - - + - + + - - + + + + - + - - - + - + + - + + - - + - - + + + - - - + - - + - + + + - + - + + - + - - - + - - + + + + - - + - + + - - + - + + - + - - + - + + - - - + - - + + - + - + + - + - - - + - + + + + - + - - - - + - + + - + + - - + + - + + - + - - + - + + - + - -

The sequence, together with the corresponding integers

1+ 2- 3+ 4+ 5- 6- 7- 8- 9+ 10+ 11- 12+ 13+ 14+ 15- 16- 17+ 18- 19+ 20+ 21- 22- 23- 24+ 25- 26+ 27+ 28- 29+ 30- 31- 32+ 33+ 34- 35+ 36- 37- 38+ 39- 40- 41+ 42+ 43- 44+ 45+ 46- 47- 48- 49+ 50+ 51+ 52- 53- 54+ 55- 56+ 57- 58- 59+ 60+ 61+ 62+ 63- 64- 65+ 66- 67- 68+ 69+ 70- 71- 72+ 73+ 74+ 75- 76- 77+ 78+ 79- 80+ 81- 82- 83+ 84- 85- 86+ 87+ 88- 89+ 90- 91- 92+ 93- 94+ 95+ 96+ 97- 98- 99+ 100- 101+ 102- 103- 104+ 105+ 106- 107- 108- 109+ 110+ 111- 112+ 113+ 114+ 115- 116- 117- 118+ 119- 120- 121+ 122- 123+ 124+ 125+ 126+ 127- 128- 129- 130- 131+ 132+ 133- 134+ 135+ 136+ 137- 138- 139+ 140- 141- 142+ 143- 144- 145+ 146- 147+ 148- 149+ 150+ 151- 152+ 153+ 154+ 155- 156- 157- 158- 159+ 160+ 161+ 162+ 163- 164- 165- 166+ 167+ 168- 169+ 170- 171- 172+ 173- 174+ 175+ 176- 177- 178+ 179- 180+ 181+ 182- 183+ 184- 185+ 186- 187+ 188+ 189- 190- 191- 192+ 193+ 194- 195+ 196+ 197- 198- 199+ 200- 201- 202+ 203- 204- 205+ 206+ 207+ 208+ 209- 210+ 211- 212- 213+ 214- 215- 216+ 217+ 218- 219- 220+ 221- 222+ 223+ 224- 225- 226- 227+ 228- 229+ 230+ 231- 232+ 233+ 234+ 235- 236- 237+ 238+ 239- 240- 241+ 242- 243- 244+ 245- 246+ 247+ 248- 249- 250+ 251- 252+ 253+ 254- 255+ 256+ 257- 258- 259+ 260- 261- 262+ 263- 264+ 265+ 266- 267- 268+ 269+ 270- 271+ 272- 273+ 274- 275- 276+ 277- 278+ 279+ 280+ 281- 282- 283+ 284- 285+ 286+ 287- 288- 289- 290- 291+ 292+ 293- 294- 295+ 296- 297+ 298+ 299- 300+ 301+ 302- 303- 304+ 305- 306+ 307+ 308- 309- 310+ 311+ 312- 313+ 314- 315- 316+ 317- 318+ 319- 320- 321+ 322+ 323- 324- 325+ 326- 327+ 328+ 329+ 330- 331+ 332- 333- 334+ 335- 336+ 337- 338- 339+ 340+ 341+ 342+ 343- 344- 345- 346+ 347+ 348- 349+ 350- 351- 352+ 353- 354+ 355+ 356- 357- 358+ 359- 360+ 361+ 362- 363+ 364+ 365- 366- 367+ 368- 369- 370+ 371- 372+ 373+ 374- 375- 376+ 377+ 378- 379- 380- 381+ 382+ 383- 384- 385+ 386- 387+ 388+ 389- 390+ 391+ 392- 393- 394+ 395- 396- 397+ 398+ 399+ 400+ 401- 402- 403- 404- 405+ 406+ 407- 408+ 409+ 410- 411+ 412+ 413- 414- 415+ 416- 417- 418+ 419+ 420- 421- 422- 423+ 424+ 425- 426+ 427+ 428- 429- 430+ 431- 432+ 433+ 434- 435+ 436- 437- 438- 439+ 440- 441+ 442+ 443- 444+ 445+ 446- 447- 448+ 449- 450- 451+ 452- 453+ 454+ 455- 456- 457+ 458+ 459- 460+ 461+ 462+ 463- 464- 465- 466+ 467- 468+ 469+ 470- 471+ 472+ 473- 474- 475+ 476- 477- 478- 479+ 480+ 481+ 482- 483- 484+ 485- 486+ 487- 488- 489+ 490+ 491+ 492- 493+ 494- 495+ 496+ 497- 498+ 499- 500- 501- 502+ 503+ 504- 505+ 506- 507+ 508+ 509- 510- 511+ 512- 513- 514+ 515- 516+ 517+ 518- 519- 520+ 521- 522+ 523- 524+ 525+ 526- 527+ 528- 529+ 530- 531- 532+ 533- 534+ 535+ 536- 537- 538+ 539- 540- 541+ 542- 543+ 544+ 545+ 546- 547- 548- 549+ 550+ 551- 552+ 553+ 554+ 555- 556+ 557- 558- 559+ 560- 561+ 562+ 563- 564- 565+ 566- 567+ 568+ 569- 570+ 571- 572- 573- 574+ 575- 576+ 577+ 578- 579+ 580+ 581- 582- 583+ 584- 585- 586- 587+ 588+ 589+ 590- 591- 592+ 593- 594+ 595+ 596- 597- 598+ 599+ 600- 601+ 602- 603+ 604+ 605- 606+ 607- 608- 609- 610+ 611- 612- 613+ 614- 615+ 616+ 617- 618- 619+ 620- 621+ 622+ 623- 624+ 625+ 626- 627- 628+ 629- 630+ 631+ 632- 633+ 634+ 635- 636- 637+ 638- 639- 640+ 641- 642+ 643+ 644- 645- 646+ 647- 648- 649+ 650- 651+ 652+ 653- 654+ 655- 656- 657+ 658+ 659- 660+ 661+ 662- 663- 664+ 665- 666- 667+ 668- 669+ 670+ 671- 672- 673+ 674- 675+ 676+ 677- 678+ 679+ 680- 681- 682+ 683- 684+ 685+ 686- 687- 688+ 689+ 690- 691+ 692- 693- 694+ 695- 696+ 697+ 698- 699- 700+ 701- 702- 703+ 704- 705+ 706+ 707- 708- 709+ 710- 711+ 712+ 713- 714+ 715+ 716- 717+ 718+ 719- 720- 721+ 722- 723+ 724+ 725- 726- 727+ 728- 729- 730+ 731- 732+ 733+ 734- 735- 736+ 737- 738+ 739- 740- 741+ 742+ 743+ 744- 745+ 746- 747- 748+ 749- 750+ 751+ 752- 753- 754+ 755- 756+ 757- 758- 759+ 760+ 761+ 762- 763- 764- 765+ 766+ 767- 768+ 769+ 770- 771- 772+ 773+ 774- 775+ 776- 777+ 778+ 779- 780- 781+ 782- 783- 784+ 785- 786- 787+ 788- 789+ 790+ 791- 792+ 793- 794- 795+ 796+ 797+ 798- 799+ 800- 801- 802+ 803- 804+ 805+ 806- 807- 808+ 809- 810+ 811- 812- 813+ 814+ 815- 816- 817+ 818- 819+ 820+ 821- 822+ 823+ 824- 825- 826+ 827+ 828- 829+ 830- 831- 832+ 833- 834- 835+ 836- 837+ 838+ 839- 840+ 841+ 842- 843- 844+ 845- 846+ 847+ 848- 849+ 850+ 851- 852- 853+ 854- 855- 856+ 857+ 858+ 859- 860- 861- 862+ 863- 864- 865+ 866- 867+ 868+ 869- 870- 871+ 872+ 873+ 874+ 875- 876+ 877- 878- 879+ 880+ 881- 882- 883+ 884- 885+ 886+ 887- 888- 889+ 890- 891- 892+ 893- 894+ 895+ 896- 897- 898+ 899- 900+ 901+ 902- 903+ 904+ 905- 906- 907+ 908- 909- 910+ 911- 912+ 913+ 914- 915- 916- 917+ 918+ 919- 920- 921+ 922+ 923- 924- 925+ 926- 927+ 928+ 929- 930+ 931+ 932- 933- 934+ 935- 936- 937+ 938- 939+ 940+ 941+ 942- 943+ 944- 945- 946+ 947- 948+ 949+ 950- 951- 952+ 953- 954+ 955+ 956- 957+ 958- 959+ 960- 961- 962+ 963- 964+ 965- 966+ 967+ 968- 969- 970+ 971- 972+ 973- 974- 975+ 976+ 977+ 978- 979+ 980- 981- 982- 983+ 984+ 985+ 986- 987- 988+ 989- 990- 991+ 992- 993+ 994+ 995- 996- 997+ 998+ 999+ 1000+ 1001- 1002+ 1003- 1004- 1005- 1006+ 1007- 1008+ 1009+ 1010- 1011+ 1012+ 1013- 1014- 1015+ 1016- 1017- 1018+ 1019+ 1020+ 1021- 1022- 1023- 1024+ 1025- 1026- 1027+ 1028- 1029+ 1030+ 1031+ 1032- 1033+ 1034- 1035+ 1036+ 1037- 1038+ 1039- 1040- 1041- 1042+ 1043- 1044- 1045+ 1046+ 1047+ 1048+ 1049- 1050- 1051+ 1052- 1053+ 1054+ 1055- 1056- 1057+ 1058- 1059+ 1060+ 1061- 1062+ 1063- 1064- 1065+ 1066- 1067+ 1068+ 1069- 1070- 1071- 1072+ 1073- 1074- 1075+ 1076+ 1077- 1078+ 1079- 1080+ 1081+ 1082- 1083+ 1084- 1085- 1086- 1087+ 1088- 1089+ 1090+ 1091+ 1092+ 1093- 1094+ 1095- 1096- 1097- 1098- 1099+ 1100- 1101+ 1102+ 1103- 1104+ 1105+ 1106- 1107- 1108+ 1109+ 1110- 1111+ 1112+ 1113- 1114+ 1115- 1116- 1117+ 1118- 1119+ 1120+ 1121- 1122+ 1123- 1124-

Links to other displays and visually displayed information about the sequence

Here is an alternate formatting. Link.

Here is the sequence divided into groups of 6. Link.

Here is a Rauzy tree that conveys information about subsequences of the sequence: Link.

Multiplicativity properties of the 1124 sequence

The 1124 sequence is not weakly multiplicative, but it does appear to be close to a weakly multiplicative sequence. That is, if you are prepared to disregard a few "errors", then the number of distinct HAP-subsequences is 6. One can use this information to identify a quasi-multiplicative sequence that is in some sense close to the 1124 sequence. Some details can be found in the sequences of comments that begin here and here.

Varying the upper and lower bounds

If N(a,b) is the maximum length of a sequence of 1s and -1s with the partial sums on its HAPs bounded below by -a and above by b, then:

N(a, b) = N(b, a)

N(0, b) = b (everything must be +1)

N(1, 1) = 11 (there are 4 such sequences: choose x(0), and use the constraints x(n)+x(2n)=0 and x(1)+...+x(2n)=0 to determine the entries up to 10; then choose x(11))

N(1, 2) = 41 (there are 4 such sequences -- example below)

0 + - - + - + + - + + - - +

- + + - + - - - + - + + - -

+ + - + - + + - - - + - + +

N(1, 3) = 83 (there are 216 such sequences -- example below)

0 - + - + - + + - + + - - +

- + + - - + - - + - + + - +

+ + - - - + + - + + - - + -

+ - - + + - - + - + + + - +

- - - + + - + - + - - + - +

+ - + + - - + - + - - - + +

N(1, 4) = 131 (there are 87144 such sequences -- example below)

0 + - - + - + + - - + -

+ + - + + + + - - - + -

- + - + + - - + + + - -

- + + - + - + - - + + -

+ + - - + - - + - + + +

+ - - + - + - - + + + -

+ - - - - + + - - - + +

- - + + + + - - - - + +

- + - + + + + - - + + -

+ - - - + + - - - + - -

+ + + - + + - - + - - +

N(2, 2) >= 1124

--Alec 13:46, 9 January 2010 (UTC)