"Squares containing at most three distinct digits"
[ Patrick De Geest ]

N.J.A. Sloane (2001), Part of the
On-Line Encyclopedia of Integer Sequences

LEGEND
References
LINK 1 : http://arxiv.org/abs/2112.00444 ( Sascha Kurz and three of his students )
LINK 2 : http://www.asahi-net.or.jp/~KC2H-MSM/mathland/overview.htm ( Hisanori Mishima's website )
TRIPLESROOTSQUARESOURCEPATTERNS & SPORADIC RECORDS (mostly L⩾17)

Four infinite patterns
1(0n)1 2 = 1(0n)2(0n)1  [n⩾0]
1(0n)11 2 = 1(0n)22(0n-1)121  [n⩾1]
11(0n)1 2 = 121(0n-1)22(0n)1  [n⩾1]
101(0n)10401(0n)101 2 =
10201(0n-2)2101002(0n-4)110221001(0n-4)2101002(0n-2)10201  [n⩾4]

[17] 10959977245460011 2 =
[33] 120121101221001210210111000120121

[18] 110000500908955011 2 =
[35] 12100110200221012201210212022010121

[20] 10099510939154979751 2 =
[39] 102000121210111101102120011101220022001

[24] 471287714788971663493899 2 =
[48] 222112110111011100020110111110102200012010222201

[29] 10000009999995510010001000001 2 =
[57] 100000200000010200110220220200010211120011021020002000001
0 1 3

( No rootsolutions less than 1024
info from Hisanori Mishima's website )

Six infinite patterns
1(0n)2 2 = 1(0n)4(0n)4  [n⩾0]
2(0n)1 2 = 4(0n)4(0n)1  [n⩾0]
102(0n)201 2 =
10404(0n-2)41004(0n-2)40401  [n⩾2]
1(0n)202(0n)1 2 =
1(0n)404(0n-2)41004(0n-2)404(0n)1  [n⩾2]
201(0n)102 2 =
40401(0n-2)41004(0n-2)10404  [n⩾2]
2(0n)101(0n)2 2 =
4(0n)404(0n-2)11001(0n-2)404(0n)4  [n⩾2]

[17] 10677612092787462 2 =
[33] 114011400004041044011001104401444
[18] 105423154192999799 2 =
[35] 11114041440001011101141014414040401
[19] 3743127183788194652 2 =
[38] 14011001114014141144414101441441401104
[22] 3180252254777039538502 2 =
[44] 10114004404014444004140001011411401140404004

[17] 23452400954944999 2 =
[33] 550015110551505101015151115110001

Three infinite patterns
1(0n)3(0n)1 2 = 1(0n)6(0n-1)11(0n)6(0n)1  [n⩾1]
1(0n)8(0n)1 2 =
1(0n-1)16(0n-1)66(0n-1)16(0n)1  [n⩾1]
4(0n)127(0n+2)4 2 =
16(0n-1)1016(0n-2)16161(0n-1)1016(0n+1)16
[n⩾2]

[17] 12649351807945204 2 =
[33] 160006101161166601100660666601616

[26] 77470059130002034719700749 2 =
[52] 6001610061606011616611060006010661000616100111161001

[16] 8427200114569499 2 =
[32] 71017701771000177071770101111001

Four infinite patterns
9(0n)1 2 = 81(0n-1)18(0n)1  [n⩾1]
1(0n+1)9 2 = 1(0n)18(0n)81  [n⩾0]
1(0n)4(0n)1 2 = 1(0n)8(0n-1)18(0n)8(0n)1  [n⩾1]
1(0n+1)9(0n)1 2 =
1(0n)18(0n-1)101(0n-1)18(0n)1  [n⩾1]

[18] 283160940117244651 2 =
[35] 80180118008081811188010188188111801
[18] 331680389653656009 2 =
[36] 110011880880801080101881180101808081
[28] 2981276371121751737986262751 2 =
[55] 8888008801008880800188080010010188818118011188010088001

[20] 43694278824566964251 2 =
[40] 1909190001999001011109190090109911991001
[27] 100990098979999970099500001 2 =
[53] 10199000091990191001091091099001091999900190199000001
0 2 3---combination impossible

One infinite pattern
2(0n)6(0n)2 2 =
4(0n-1)24(0n-1)44(0n-1)24(0n)4  [n⩾1]

[16] 1562062816343832 2 =
[31] 2440040242204024220420044444224

[24] 205524700326856587391168 2 =
[47] 42240402444444204240022400420200244000244404224

[27] 634050802727999251005000002 2 =
[54] 402020420440020222440222024020044204422022004020000004

Four infinite patterns
5(0n)5 2 = 25(0n-1)5(0n)25  [n⩾1]
5(0n+1)505 2 = 25(0n)505(0n-1)255025  [n⩾1]
505(0n+2)5 2 = 255025(0n-1)505(0n+2)25  [n⩾1]
15(0n+2)85(0n)15 2 =
225(0n)255(0n)52225(0n-2)255(0n)225  [n⩾2]
Farideh Firoozbakht's intricate patterns

[17] 14899671139156245 2 =
[33] 222000200055005555555250522500025

[17] 50000050049995505 2 =
[34] 2500005005002055502050050520205025

[20] 23500043724538482665 2 =
[39] 552252055055220520520522052500505502225

[20] 23558545158870178045 2 =
[39] 555005050002525502502022522050000022025

[21] 150167406766664999985 2 =
[41] 22550250055025025225250200022000050000225

[22] 5000000500000500254955 2 =
[44] 25000005000005252550050255205255020002052025

[22] 5000005004997549999955 2 =
[44] 25000050050000550000025505552050220500002025

[23] 44949994999999949999995 2 =
[46] 2020502050500020505000050500052500000500000025

[24] 447241797269721007814765 2 =
[48] 200025225225050225520202505522022522200552005225

[25] 5000000500499975499999955 2 =
[50] 25000005005000005500225025500555205002205000002025

[27] 500000000000500000500254955 2 =
[54] 250000000000500000500255205000500255205255020002052025

[29] 50000005004999999999955050005 2 =
[58] 2500000500500025050020505000050050550050002020502050500025

[29] 50050000004999999999955050005 2 =
[58] 2505002500500500000020500505500050500050002020502050500025

[30] 141422082876067219949805050005 2 =
[59] 20000205525005225202000505202222520222202205205000550500025
0 2 6---combination impossible
0 2 7---combination impossible
0 2 8---combination impossible

[17] 96385117673990673 2 =
[34] 9290090909029029202290909290992929
[18] 151400161921673747 2 =
[35] 22922009029909029220029029909020009
[21] 149067065510873088673 2 =
[41] 22220990020022929092929022220290920900929

[19] 5773251280200207952 2 =
[38] 33330430344333340030340440344044034304
[23] 20832739723817975138362 2 =
[45] 434003044400343443044430000434430333044043044
0 3 5---combination impossible

[17] 25107103902348156 2 =
[33] 630366666363306003336330636600336
0 3 7---combination impossible
0 3 8---combination impossible

[9] 969071253 2 =
[18] 939099093390990009

[20] 21082112192576920612 2 =
[39] 444455454500400455000455440400550454544
[20] 63639964217112858462 2 =
[40] 4050045045555405040055544045540445005444
[22] 6674983479713230005962 2 =
[44] 44555404454444540454555540045000554555545444
[26] 21214250022106461574572502 2 =
[51] 450044404000444005405445044455050544404404054540004
[28] 2108436491907081488939581538 2 =
[55] 4445504440405440505004450045555054500055550554550445444

One infinite pattern
8(0n)254(0n+2)8 2 =
64(0n-1)4064(0n-2)64644(0n-1)4064(0n+1)64  [n⩾2]

[18] 635613598504629262 2 =
[36] 404004646604004046006460044066664644
[19] 2457776365832743262 2 =
[37] 6040664664446006640606666600406400644
[19] 2542962918459579238 2 =
[37] 6466660404660460644444606044000660644

[16] 2010988315424552 2 =
[31] 4044074004774077447400004400704

Five infinite patterns
2(0n)2 2 = 4(0n)8(0n)4  [n⩾0]
2(0n)2(0n-1)2 2 =
4(0n)8(0n-1)84(0n-1)8(0n-1)4  [n⩾1]
2(0n-1)2(0n)2 2 =
4(0n-1)8(0n-1)48(0n-1)8(0n)4  [n⩾1]
2(0n)22 2 =
4(0n)88(0n-1)484  [n⩾1]
22(0n)2 2 =
484(0n-1)88(0n)4  [n⩾1]

[17] 21919954490920022 2 =
[33] 480484404884004840840444000480484
[18] 220001001817910022 2 =
[35] 48400440800884048804840848088040484
[20] 20199021878309959502 2 =
[39] 408000484840444404408480044404880088004
[24] 942575429577943326987798 2 =
[48] 888448440444044400080440444440408800048040888804
[29] 20000019999991020020002000002 2 =
[57] 400000800000040800440880880800040844480044084080008000004

Two infinite patterns
(9n)7 2 = (9n)4(0n)9  [n⩾1]
2(0n)1(0n)2 2 = 4(0n)4(0n)9(0n)4(0n)4  [n⩾0]
[17] 99704560597822753 2 =
[34] 9940999404004909449099404004499009
[22] 3015775265159011230138 2 =
[43] 9094900449944904494440090444449999999499044

[16] 2236081408416666 2 =
[31] 5000060065066660656065066555556
0 5 7---combination impossible
0 5 8---combination impossible

[11] 70778174997 2 =
[22] 5009550055905955950009
[12] 771395165003 2 =
[24] 595050500590005595990009

[20] 26012881552428213576 2 =
[39] 676670006660660066767076066770670707776
0 6 8---combination impossible

[17] 30000101109940614 2 =
[33] 900006066606660060090966606696996
[29] 24691314243454114014126412353 2 =
[57] 609660999069000006699096996606066090996096009966990996609
0 7 8---combination impossible

[16] 8819172285373497 2 =
[32] 77777799799099990007000790009009

[15] 301345331969667 2 =
[29] 90809009099908808089808090889
[27] 299831600904572582192518303 2 =
[53] 89898988900998890088080098089989880890999988989999809

[20] 56843832676142723489 2 =
[40] 3231221313313311221231322223122312333121
[21] 557963558954625926861 2 =
[42] 311323333121312322332133323111223321313321

One infinite pattern
(3n)8 2 = (1n)4(2n-1)44  [n⩾1]
[16] 4705573731461671 2 =
[32] 22142424142222114221142142112241
[21] 379766258564954821662 2 =
[42] 144222411144424121442444112111142224442244
[28] 1114110597927523626433041668 2 =
[55] 1241242424414424212214142144114212224412421422224222224

Three infinite patterns
(3n)5 2 = (1n)(2n+1)5  [n⩾0]
123(3n)5 2 = 152(1n)5(2n+2)5  [n⩾0]
(3n)504485 2 =
(1n)2251(2n-4)51515115225  [n⩾4]

[18] 159722338802442489 2 =
[35] 25511225512522225551152512152515121
[21] 124588166420914599285 2 =
[41] 15522211212125512112255222511252122511225
[21] 150051695267169681235 2 =
[41] 22515511252551552115222525221211511125225
[23] 34816980372445012123335 2 =
[46] 1212222122255221215111111512151155125251522225
[25] 1102340925268369741032335 2 =
[49] 1215155515521525522215211555521512552151515552225
[28] 4638162516046117503822620335 2 =
[56] 21512551525255251211125255525551555121211151225555512225

[20] 47130268582155593596 2 =
[40] 2221262216626122626662262611111116211216
[26] 51595698572871617009432954 2 =
[52] 2662116111222626216162621266666216222222216621166116

[18] 130834904430015239 2 =
[35] 17117772217211221211117217772227121

One infinite pattern
(3n)59 2 = (1n)28(2n-1)881  [n⩾1]
[17] 28616100692061141 2 =
[33] 818881218818182112888822882221881
[17] 33350084135098989 2 =
[34] 1112228111818181281181888828822121
[19] 1490909832561385391 2 =
[37] 2222812128828218222281282181228222881
[20] 34377642169166984891 2 =
[40] 1181822281111288118221882821111822281881

[17] 99961098929489127 2 =
[34] 9992221299191112291912929211222129
[18] 459556524411439511 2 =
[36] 211192199129121999192121999211919121
[27] 145303131149776986249167839 2 =
[53] 21112999921929291129929211912291229929191119991929921

[20] 21079405433537116521 2 =
[39] 444341333431434111311131411343131143441
[30] 177324875114669443080086908188 2 =
[59] 31444111334433114334141133143444444313434111431113141443344
1 3 5---impossible since all 3 digits odd

[17] 18266544874814631 2 =
[33] 333666661663616663311166611666161
[29] 11537606482136410218512760694 2 =
[57] 133116363336636111166666336113333136136363666113311361636
1 3 7---impossible since all 3 digits odd

[24] 286074095527510693610891 2 =
[47] 81838388131883313833383381811133318888113813881
1 3 9---impossible since all 3 digits odd

[14] 73560479506012 2 =
[28] 5411144145154411455544144144

[19] 1290017214657004546 2 =
[37] 1664144414111416144464164661464666116
[21] 375720828696801774892 2 =
[42] 141166141116611464114116414644641441611664
[21] 802565644925350914229 2 =
[42] 644111614414444441664646611416446114664441
[22] 4051067284576580130696 2 =
[44] 16411146144166666464466464146441416441444416
[23] 10789398111648380852704 2 =
[45] 116411111611641646616166611414441166144111616
[30] 105567345643273687982611367608 2 =
[59] 11144464466166416111166414141141166144146161144464111641664

[19] 1071033028175028538 2 =
[37] 1147111747441771474111741117114417444
[22] 2177492084289725902412 2 =
[43] 4741471777144414774147714111744447747417744

[18] 284800189808379191 2 =
[35] 81111148114888814414411114441814481
[19] 2117642891144336478 2 =
[37] 4484411414414144114118188814881444484
[19] 6435147099182306059 2 =
[38] 41411118188114448414441181181148111481
[25] 1346900557360669225841779 2 =
[49] 1814141111418481411488184148411114184811141884841
[28] 2209483759119790145920022988 2 =
[55] 4881818481814118844811411488844844184118148818448448144
1 4 9A027675A006716 *
(* seq by
Neil Sloane)

[18] 648070211589107021 2 =
[36] 419994999149149944149149944191494441

One infinite pattern
(3n)4 2 = (1n+1)(5n)6  [n⩾0]
[17] 12472031176057954 2 =
[33] 155551561656561551165551166666116
[19] 2482270463831785216 2 =
[37] 6161666655611666116165616661556166656
[20] 12516036335682176169 2 =
[39] 156651165556116515661615565555551516561
[21] 258100003219026842869 2 =
[41] 66615611661661666651111615111161616151161
[23] 74240565428619726479296 2 =
[46] 5511661555161166511651661611666516615516655616
1 5 7---impossible since all 3 digits odd

[19] 3399291958357679641 2 =
[38] 11555185818155188818511188881585888881
[22] 2412237158970509643109 2 =
[43] 5818888111118115811551585855811558551185881
1 5 9---impossible since all 3 digits odd

[10] 1292931424 2 =
[19] 1671671667166667776

[17] 12723468913060546 2 =
[33] 161886661181618111866616661818116
[18] 126931550889393381 2 =
[35] 16111618611186661611161686166611161
[21] 431495267861269619604 2 =
[42] 186188166186668818681818186188818861116816
[25] 1080794204132598414568541 2 =
[49] 1168116111686616811868118886618818666111186868681

[19] 3019927482025216937 2 =
[37] 9119961996691166966116961161911661969
[19] 4082512947923373236 2 =
[38] 16666911969961991191619191116961111696
[21] 411900436901564744737 2 =
[42] 169661969919699919691616999616691969199169
[23] 34149670012924966713187 2 =
[46] 1166199961991666696199696161116991961919696969
[30] 248216864092061020657513399437 2 =
[59] 61611611619696691696969619616966119999999161919199911916969

[9] 105769141 2 =
[17] 11187111187877881
[9] 279067891 2 =
[17] 77878887787187881
1 7 9---impossible since all 3 digits odd

[16] 2969848344609859 2 =
[31] 8819999189981919818818919999881

[15] 205483392086668 2 =
[29] 42223424423443333243223342224

[18] 159789024443333515 2 =
[35] 25532532332552235533223325522255225
[27] 576387476638096486959455635 2 =
[54] 332222523225232223533222222253253255253352335533253225

[16] 2514602599284156 2 =
[31] 6323226232326633633323632632336
[27] 251462176552105392823457806 2 =
[53] 63233226236322223226263332266636366232262662262333636
2 3 7---combination impossible
2 3 8---combination impossible

[19] 1814641285211195673 2 =
[37] 3292922993992939999922923222293922929
[20] 14940646884386874573 2 =
[39] 223222929323939222223332232999233932329

Three infinite patterns
(6n)5 2 = (4n)(2n+1)5  [n⩾0]
(6n)515 2 = (4n)24(2n-1)45225  [n⩾1]
2(3n)5 2 = 5(4n-1)5(2n+1)5  [n⩾1]
[22] 1566985170463509665838 2 =
[43] 2455442524452554445254444455245254424242244
[22] 5022405325580985229335 2 =
[44] 25224555254424242244454444254455442544542225
[22] 5042276720337304556485 2 =
[44] 25424554524455524225542255422225542555555225
[24] 156602211177063382566485 2 =
[47] 24524252545545555425452252442555424225445255225
[24] 473545618135383472428338 2 =
[48] 224245452455222424254455242452522555442545442244
[27] 212706944938912242495946332 2 =
[53] 45244244425245444452555552444225545225555452224254224

One infinite pattern
(6n)8 2 = (4n)6(2n)4  [n⩾0]
[17] 14988743331128338 2 =
[33] 224662426646444226244244226642244
[18] 211335426908131668 2 =
[35] 44662662666442262666444262424462224
[19] 4964540927663570432 2 =
[38] 24646666622446664464662466646224666624
[22] 8015264452445860907338 2 =
[44] 64244464242642246466444266646244264622246244
[25] 1562826497869300353470568 2 =
[49] 2442426662442422262266222264624646466242442242624
[26] 47144739098275455895604568 2 =
[52] 2222626424644462446266642442624422642664422222466624
[27] 163176169897520398456349838 2 =
[53] 26626462422424442244464226222466224666246222642626244
[30] 162062538622046218465618335432 2 =
[59] 26264266424622222222622644266266266222662426642466466626624

[11] 88002411582 2 =
[22] 7744424444247727742724
[17] 47146022358675418 2 =
[34] 2222747424244722424422447477474724

[18] 669644852476481662 2 =
[36] 448424228448248888288442822222282244
[20] 20597146608802018338 2 =
[39] 424242448424484484244824228422488282244
[23] 53710727465081333454522 2 =
[46] 2884842244828242284244242842824284482242248484
[25] 2069416058768323727702022 2 =
[49] 4282482824288222284288288882288482848444822888484
[27] 149140498954591218312271662 2 =
[53] 22242888428424424282282224442842448448442222888242244
[29] 20598117118436403877526792022 2 =
[58] 424282428824822822284282828848288484482824448422442848484

[21] 222468490448488507807 2 =
[41] 49492229242429222429494944494949499949249
[21] 547259530974381470838 2 =
[42] 299492994242299992492444422992424244422244

[18] 237112320688458875 2 =
[35] 56222252622266562625655622566265625
[18] 257395128676171075 2 =
[35] 66252252266222665256252522666655625
[18] 745355990494250516 2 =
[36] 555555552565665265566665252566266256
[19] 7453363177489241484 2 =
[38] 55552622655552522252252226565666522256
[20] 80781591501545428925 2 =
[40] 6525665525522556666225656625562226655625
[21] 228618071522411733125 2 =
[41] 52266222626626566662522622656666222265625
[24] 237750344316525128700475 2 =
[47] 56525226222626252566552565525652262562265225625
[24] 516291221749568609228465 2 =
[48] 266556625655662226525525225552225265562566256225
[26] 81401637345465395512991484 2 =
[52] 6626226562522666562566262626266252566552622656522256
[29] 14920674457351323857264196585 2 =
[57] 222626526262256222655556652225555252265566656525525662225
[29] 23508012597117321085533117075 2 =
[57] 552626656266226655522226225556662256522626266565656555625

One infinite pattern
1(6n)5 2 = 2(7n)(2n+1)5  [n⩾0]
[18] 870185357137045415 2 =
[36] 757222555775727275772757755772522225
[26] 52174924557278712520943915 2 =
[52] 2722222752557725255755757775775772222257522575527225

[18] 159013392166264585 2 =
[35] 25285258888222255288288552225222225

[17] 77170285247817565 2 =
[34] 5955252925229529299525595522529225
[19] 9794365502654705173 2 =
[38] 95929595599592555525252922555552959929
[22] 1598601020441309256565 2 =
[43] 2555525222555995255555255252529952995599225
[26] 30484348812550551609088485 2 =
[51] 929295522525252225925225599299529559999252559595225
[26] 76975943837016723668817565 2 =
[52] 5925295929599552922952299222959292529595925252529225

[18] 150573163864701424 2 =
[35] 22672277676226226672276776667627776
2 6 8---combination impossible

[20] 78883604126137785577 2 =
[40] 6222622999929222269696699966269229222929
[23] 47567102808870567435673 2 =
[46] 2262629269629662226292666922999622262992962929
[26] 26395073915340646948470264 2 =
[51] 696699926996296229992699669262629222696929692229696
2 7 8---combination impossible

[11] 14907304327 2 =
[21] 222227722297792922929
[27] 850281851974525726895170673 2 =
[54] 722979227797229279772792797979272222799797729799272929

[13] 5405829167667 2 =
[26] 29222988989999289998222889
[26] 17320185602062360469701767 2 =
[51] 299988829289888292222928892882898892999989922922289

[20] 21343231796858797962 2 =
[39] 455533543534444433554534333443535353444

[17] 66817996351009092 2 =
[34] 4464644636363464333646646666664464
[19] 6666680833328031344 2 =
[38] 44444633333463334436443663666646446336

[17] 21150351639576462 2 =
[33] 447337374477734734334374744437444
[20] 20913496712251033188 2 =
[39] 437374344733334774447744773373477443344
[22] 1864878916830083039312 2 =
[43] 3477773374437343773773333743737447337433344
[27] 271006150065722262703289312 2 =
[53] 73444333373444774773333473377377773344743344373433344

[19] 6952948212333013522 2 =
[38] 48343488843384848488834343333834844484
[25] 5906300402396058566810062 2 =
[50] 34884384443343843348888488484833488344838384443844

[21] 185605616817891584607 2 =
[41] 34449444994349999433343349994949439344449

[17] 23527926717739784 2 =
[33] 553563335635333565566365536366656
3 5 7---impossible since all 3 digits odd
3 5 8---combination impossible
3 5 9---impossible since all 3 digits odd

[17] 25226323103806424 2 =
[33] 636367377337637773373677663667776
[19] 1932967502917049474 2 =
[37] 3736363367333373666777367633763676676

[15] 183539278812156 2 =
[29] 33686666866886336386333368336

[23] 18430047920827535573187 2 =
[45] 339666666363999366939366969366969936633336969
[26] 31047286456844613647179386 2 =
[51] 963933996333366963633963999333393696336393663336996
3 7 8---combination impossible
3 7 9---impossible since all 3 digits odd

[12] 199974958167 2 =
[23] 39989983893893399999889

[17] 25425667278648884 2 =
[33] 646464556564556546656456554445456
[21] 237605691124298293112 2 =
[41] 56456464454655444665444666454556666644544
[23] 75269219840819770294592 2 =
[46] 5665455455445656566644565645546455454464446464
[24] 675754811056988742949784 2 =
[48] 456644564666666555445565455644644555565545646656
[26] 25781108305591628417975738 2 =
[51] 664665545464645645665646644665564654546645556644644

[16] 8629863583949388 2 =
[32] 74474545477575775744575745574544
[29] 27341447393189418631675507588 2 =
[57] 747554745554544455555455747745774457774554454557445577744

[15] 767175898056538 2 =
[30] 588558858558855585845444545444
[30] 293061185503724726684202222622 2 =
[59] 85884858448848555885485448555548484845848584584884848554884

[24] 703957001491895099962643 2 =
[48] 495555459949459999994555545994544549499995545449

[18] 253863101778786762 2 =
[35] 64446474444746646444646744666444644
[24] 815277035409723028858892 2 =
[48] 664676644466466777446466766667744446667647467664

[20] 26204321529981784378 2 =
[39] 686666466846666884868486448488884846884

[21] 999997323321167445187 2 =
[42] 999994646649499499946646996649944649464969
[23] 22236561446627600040614 2 =
[45] 494464664969644944649644494946666694449496996
[25] 9718263579193026119075264 2 =
[50] 94444646994669646646646969664664969996646496669696

[19] 9208064263934568938 2 =
[38] 84788447488748874848488744487874447844
[20] 22019193553462506122 2 =
[39] 484844884744844787448744774844887478884

[10] 8819171038 2 =
[20] 77777777797497997444
[27] 865996535661545126193725357 2 =
[54] 749949999777797799497449749797947997449449477944777449

One infinite pattern
(6n)7 2 = (4n+1)(8n)9  [n⩾0]
[10] 6670081667 2 =
[20] 44489989444449498889
[26] 30732718558321504090886022 2 =
[51] 944499989984998988844994898844848999494444994984484

[18] 815816631091556424 2 =
[36] 665556775565576667756557666775667776
[21] 881854102200458483334 2 =
[42] 777666657567776675657756675676567555755556
[22] 2562353735836753390526 2 =
[43] 6565656667556566576676575665776756666556676

Only this one rather trivial solution is known
[3] 816 2 =
[6] 665856

[18] 834023722663550236 2 =
[36] 695595569965566559565695699695655696
5 7 8---combination impossible
5 7 9---impossible since all 3 digits odd

[11] 92496431583 2 =
[22] 8555589855588599885889

( No rootsolutions under 1025
info from Hisanori Mishima's website )

[16] 9831977256725526 2 =
[32] 96667776776767999797799699976676

[22] 2588184048685235767383 2 =
[43] 6698696669868698868988986669668968886668689

[16] 9949370777987917 2 =
[32] 98989978877879888789778997998889

```

```

Farideh Firoozbakht (email) (1962–2019 †)
A058426 (squares with digits 0,2 & 5)
[ di 9/12/2008 20:37 ]

I found the following interesting infinite pattern for numbers m such that
the digits of m2 are 0, 2 & 5 which isn't in the mentioned in the table

a(n) = 4.4.(92^1–1).4.(92^2–1).4.(92^3–1). ... .4.(92^n–1).5      [ n > 0 ]

b(n) = a(n)2 = {20} . {20.5.(02^1–1)} . {20.5.(02^1–1).5.(02^2–1)} . ... . {20.5.(02^1–1).5.(02^2–1) . ... . 5.(02^n–1)} . {25}

Examples :

a(1) = 4.4.9.5 = 4495
b(1) = {20}.{2050}.{25} = 20205025

a(2) = 4.4.9.4.999.5 = 44949995
b(2) = {20}.{2050}.{20505000}.{25} =2020502050500025

a(3) = 4.4.9.4.999.4.9999999.5 = 4494999499999995
b(3) = {20}.{2050}.{20505000}.{2050500050000000}.{25} = 20205020505000205050005000000025

a(4) = 4.4.9.4.999.4.9999999.4.999999999999999.5 = 44949994999999949999999999999995
b(4) = {20}.{2050}.{20505000}.{2050500050000000}. {20505000500000005000000000000000}.{25}
= 2020502050500020505000500000002050500050000000500000000000000025

I hope that the pattern and examples are clear.

Best wishes,
Farideh

Farideh Firoozbakht (email) (1962–2019 †)
A058426 (squares with digits 0,2 & 5)
[ di 16/12/2008 17:22 ]

I also found the following similar patterns a(n, k) for [ 1 < k < n+1 ].

a(n, k) = 4.4.(92^1–1).4.(92^2–1).4.(92^3–1). ... .4.(92^n–1).4.(92^k–2).5      [1 < k < n+1]

The formula for b(n, k) = a(n, k)2 is more intricate than of b(n).

I wrote the formula of b(n, 2) after the following examples :

a(2, 2) = 4.4.9.4.999.4.99.5 = 44949994995
b(2, 2) = 2020502050050525050025

a(3, 2) = 4.4.9.4.999.4.9999999.4.99.5 = 4494999499999994995
b(3, 2) = 20205020505000205005055005000025050025

a(3, 3) = 4.4.9.4.999.4.9999999.4.999999.5 = 44949994999999949999995
b(3, 3) = 2020502050500020505000050500052500000500000025

a(4, 2) = 4.4.9.4.999.4.9999999.4.9999999999999994.99.5
= 44949994999999949999999999999994995
b(4, 2) = 2020502050500020505000500000002050050550050000500500000000000025050025

a(4, 3) = 4.4.9.4.999.4.9999999.4.9999999999999994.999999.5
= 449499949999999499999999999999949999995
b(4, 3) = 202050205050002050500050000000205050000505000550000005000000002500000500000025

a(4, 4) = 4.4.9.4.999.4.9999999.4.9999999999999994.99999999999999.5
= 44949994999999949999999999999994999999999999995
b(4, 4) = 2020502050500020505000500000002050500050000000050500050000000525000000000000050000000000000025

b(n, 2) for [ n > 2 ], includes 2n substrings and I separated them with "." and put them
between parentheses as follows.

b(n, 2) =

{20205}.
{(02^1–1).205}.
{(02^1–1).5.(02^2–1).205}.
{(02^1–1).5.(02^2
1).5.(02^3–1).205}.
... .
{(02^1–1).5.(02^2–1).5.(02^3–1). ... .5.(02^(n–1)–1).205}.
{00505}.
{5005.(02^3–4)}.
{5005.(02^4–4)}.
...
{5005.(02^n–4)}.
{25050025}

Examples :

b(3, 2) = a(3, 2)2 =

{20205}.
{(02^1–1).205}.
{(02^1–1).5.(02^2–1).205}.
{00505}.
{5005.(02^3–4)}.
{25050025}

=
20205.
0205.
0.5.000.205.
00505.
5005.0000.
25050025.

= 20205020505000205005055005000025050025

b(4, 2) = a(4, 2)2 =

{20205}.
{(02^1–1).205}.
{(02^1–1).5.(02^2–1).205}.
{(02^1–1).5.(02^2–1).5.(02^3–1).205}.
{00505}.
{5005.(02^3–4)}.
{5005.(02^4–4)}.
{25050025}

=
20205.
0205.
0.5.000.205.
0.5.000.5.0000000.205.
00505.
5005.0000.
5005.000000000000.
25050025

= 2020502050500020505000500000002050050550050000500500000000000025050025

Hope that the formula for b(n,2) and the examples are clear.

Best wishes,
Farideh

Farideh Firoozbakht (email) (1962–2019 †)
A058426 (squares with digits 0,2 & 5)
[ vr 19/12/2008 16:22 ]

I also found a very nice formula for b(n, k) :

b(n, k), [ 1 < k < n ] :

b(n, k) =

{20}.

{20.5.(02^1–1)}.
{20.5.(02^1–1).5.(02^2–1)}.
{20.5.(02^1–1).5.(02^2–1).5.(02^3–1)}.
{20.5.(02^1–1).5.(02^2–1).5.(02^3–1).5.(02^4–1)}.
{20.5.(02^1–1).5.(02^2–1).5.(02^3–1).5.(02^4–1).5.(02^5–1)}.
...
{20.5.(02^1–1).5.(02^2–1).5.(02^3–1).5.(02^4–1). ... .5.(02^(n–1)–1)}.

{20.5.(02^1–1).5.(02^2–1).5.(02^3–1). ... .5.(02^(k–2)–1).5.(02^(k–1)).
5.(02^1–1).5.(02^2–1).5.(02^3–1). ... .5.(02^(k–2)–1).5.(02^(k–1)–1).5}.

{5.(02^k–2).5.(02^(k+1)–2^k)}.
{5.(02^k–2).5.(02^(k+2)–2^k)}.
...
{5.(02^k–2).5.(02^n–2^k)}.

{25.(02^k–3).5.(02^k–2).25}

and

b(n, n), [ n > 2 ] :

b(n, n) =

{20}.

{20.5.(02^1–1)}.
{20.5.(02^1–1).5.(02^2–1)}.
{20.5.(02^1–1).5.(02^2–1).5.(02^3–1)}.
{20.5.(02^1–1).5.(02^2–1).5.(02^3–1).5.(02^4–1)}.
{20.5.(02^1–1).5.(02^2–1).5.(02^3–1).5.(02^4–1).5.(02^5–1)}.
...
{20.5.(02^1–1).5.(02^2–1).5.(02^3–1).5.(02^4–1). ... .5.(02^(n–1)–1)}.

{20.5.(02^1–1).5.(02^2–1).5.(02^3–1). ... .5.(02^(n–2)–1).5.(02^(n–1)).
5.(02^1–1).5.(02^2–1).5.(02^3–1). ... .5.(02^(n–2)–1).5.(02^(n–1)–1).5}.

{25.(02^n–3).5.(02^n–2).25}

Dear Farideh,

What intricate and beautiful patterns you discovered there! Thanks a lot !

Farideh Firoozbakht (email) (1962–2019 †)
A058426 (squares with digits 0,2 & 5)
[ di 24/02/2009 9:24 ]

Since the following six numbers are solutions of infinite patterns so they
must not be in the table of sporadic solutions.

1. 4495
2. 4949995
3. 4949994995
4. 4494999499499995
5. 4494999499999995
6. 4494999499999994995

The number 44949994999999949999995 isn't a solution of my infinite pattern.

Patrick replied :
But are you sure that the last number couldn't be made part of that
infinite pattern as well. It does look so similar to the others, perhaps
an adaptation 'somehow' of your pattern could include this one as well ?

Farideh Firoozbakht (email) (1962–2019 †)
A058426 (squares with digits 0,2 & 5) - New patterns
[ di 26/02/2009 4:29 ]

1.
For each natural number n, [ n > 1 ] there exist n numbers m of the following
form where m2 has only three distinct digits 0, 2 & 5.

44.(92^1–1).4(92^2–1).4. ... (92^n–1).4.99.4.(9k).5

Where k is in the set A(n) = {2^3–2, 2^4–2, ..., 2^n–2, 2^(n+1)–4, 2^(n+1)+2}.

Note that A(n) has n elements and A(2) has only the two last terms.

Since for [ n > 2 ] we have min(A(n)) = 6 and only for [ n = 2 ] min(A(n)) = 4,
I considered the two solutions related to A(2) = {4,10} namely 4494999499499995 &

But as you expected we can include these two solutions, specially the first one
which is correspondent to 4 ( the smallest term of A(2) ), in these infinite patterns.

Suppose that the i-th term (in increasing order) of A(n) = A(n,i) now we define :

c(n,i) = 44.(92^1–1).4(92^2–1).4. ... (92^n–1).4.99.4.(9A(n,i)).5
and
cc(n,i) = c(n,i)2.

Examples :

c(2,1) = 44.(92^1–1).4.(92^2–1).4.99.4.(9A(2,1)).5

c(2,2) = 44.(92^1–1).4.(92^2–1).4.99.4.(9A(2,2)).5

c(3,2) = 44.(92^1–1).4.(92^2–1).4.(92^3–1).4.99.4.(9A(3,2)).5

A(2,1) = 2^(2+2)–4 = 4 ; A(2,2) = 2^(2+2)+2 = 10 ; A(3,2) = 2^(3+1)–4 = 12

Hence,

c(2,1) = 4494999499499995
c(2,2) = 4494999499499999999995
c(3,2) = 44949994999999949949999999999995

cc(2,1) = 20205020500505205550255005000025
cc(2,2) = 20205020500505250500205050005005000000000025
cc(3,2) = 2020502050500020500505500500002055502550000000500500000000000025

2.
For each natural number n, there exists a number d(n),

d(n) = 5.(03*2^(n–1)–1).5.(03*2^(n–2)–1). ... . 5.(03*2^0–1).49995505,

where dd(n) = d(n)2 has only three distinct digits 0, 2 & 5.

Note that we have d(n+1) = 5.(03*2^n–1).d(n) and number of digits of d(n) equals to
8 + (3*2^0 + 3*2^1 + ... + 3*2^(n-1)) = 3*2^n + 5.

So for the sequence {d(n)} we obtain the following recursion relation.

d(1) = 50049995505, d(n+1) = d(n) + 5*10^(3*2^(n+1)+4).

Examples :

d(2) = 5.(03*2^1–1).5.(03*2^0–1).49995505 = 50000050049995505
dd(2) = 2500005005002055502050050520205025

d(3) = 5.(03*2^2–1).5.(03*2^1–1).5.(03*2^0–1).49995505
d(3) = 50000000000050000050049995505
dd(3) = 2500000000005000005005002050505005002055502050050520205025

d(4) = d(3) + 5*10^(3*2^4+4) = 50000000000050000050049995505 + 5*10^52
d(4) = 50000000000000000000000050000000000050000050049995505.

Anne Zahn (email)
Reporting an error within a pattern of 0 1 8 .
[ di 12/6/2021 20:37 ]

Hello,
I found a mistake in the table on this webpage. One of the infinite patterns is wrong.
For triple 0 1 8 there is the pattern 1(0n)4(0n)1 2 = 1(0n)818(0n)8(0n)1 [n>=1] which is not right.
It should be 1(0n)8(0n-1)18(0n)8(0n)1 .

Zhao Hui Du (email)
Update for squares containing at most three distinct digits
[ vr 1-2/3/2024 7:00 ]

Hi Patrick,
I have searched for the problem of squares containing at most three distinct digits
and found all results with length of base less than 27 digits.
The results are available at a Chinese BBS (see LINK 10).

I think we should thank the development of the IC industry for the achievement.
C++ code is used with 'openmp' to run in a machine with 40 cores to process one pattern
(square with given 3 digits) no more than 27 digits for its bases (so squares are no more than 54 digits)
The code first enumerates all integers so that the last 16 digits of its squares are in the pattern.
Next it enumerates squares whose first 16 digits are in the pattern and find square root of it.
Some digits of the two lists of integers should be overlapped for target integers so that we could
search in the first list for those integers with overlapped digits and verify the result.
The corresponding C++ code is in the BBS too.

Zhao Hui Du (email)
Second Update
[ wo 6/3/2024 23:52 ]

All results no more than 30 digits are available now such as
[30] 177324875114669443080086908188^2 = [59] 31444111334433114334141133143444444313434111431113141443344

```

```