Sloane's On-Line Encyclopedia of Integer Sequences


"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
 palegreen bgcolor indicating finished sequence
 no bgcolor indicating that sequence needs more terms
References
LINK 1 : http://www.asahi-net.or.jp/~KC2H-MSM/mathland/overview.htm
Hisanori Mishima's website
LINK 2 : http://rec-puzzles.org/sol.pl/arithmetic/digits/squares/three.digits
LINK 3 : http://www.stetson.edu/~efriedma/mathmagic/0999.html
LINK 4 : http://www.immortaltheory.com/NumberTheory/TriDigitalSquares.htm
LINK 5 : http://www.worldofnumbers.com/crump.htm
LINK 6 : http://www.mathematik.uni-bielefeld.de/~sillke/SEQUENCES/series002
LINK 7 : http://blue.kakiko.com/mmrmmr/htm/eqtn06.html
TRIPLESROOTSQUARELINKSPATTERNS & SPORADIC RECORDS
0 1 2A058411A058412SourceThree 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]
10099510939154979751 2 =
102000121210111101102120011101220022001
0 1 3  Source( next term > 10100003013030333111001 * 10 23
info from Hisanori Mishima's website )
0 1 4A058413A058414SourceSix 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]
3743127183788194652 2 =
14011001114014141144414101441441401104
0 1 5A058415A058416Source23452400954944999 2 =
550015110551505101015151115110001
0 1 6A058417A058418SourceThree 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]
12649351807945204 2 =
160006101161166601100660666601616
0 1 7A058419A058420Source8427200114569499 2 =
71017701771000177071770101111001
0 1 8A058421A058422SourceFour 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)818(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]
331680389653656009 2 =
110011880880801080101881180101808081
0 1 9A058473A058474Source43694278824566964251 2 =
1909190001999001011109190090109911991001
0 2 3---combination impossible
0 2 4A058423A058424Source One infinite pattern
2(0n)6(0n)2 2 =
4(0n-1)24(0n-1)44(0n-1)24(0n)4  [n>=1]
1562062816343832 2 =
2440040242204024220420044444224
0 2 5A058425A058426SourceFour 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]
23558545158870178045 2 =
555005050002525502502022522050000022025
0 2 6---combination impossible
0 2 7---combination impossible
0 2 8---combination impossible
0 2 9A058427A058428Source151400161921673747 2 =
22922009029909029220029029909020009
0 3 4A058429A058430Source5773251280200207952 2 =
33330430344333340030340440344044034304
0 3 5---combination impossible
0 3 6A058431A058432Source25107103902348156 2 =
630366666363306003336330636600336
0 3 7---combination impossible
0 3 8---combination impossible
0 3 9A058433A058434Source969071253 2 =
939099093390990009
0 4 5A058435A058436Source63639964217112858462 2 =
4050045045555405040055544045540445005444
0 4 6A058437A058438SourceOne infinite pattern
8(0n)254(0n+2)8 2 =
64(0n-1)4064(0n-2)64644(0n-1)4064(0n+1)64
[n>=2]
2542962918459579238 2 =
6466660404660460644444606044000660644
0 4 7A058439A058440Source2010988315424552 2 =
4044074004774077447400004400704
0 4 8A058441A058442SourceFive 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]
20199021878309959502 2 =
408000484840444404408480044404880088004
0 4 9A058443A058444SourceTwo 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]
99704560597822753 2 =
9940999404004909449099404004499009
0 5 6A058445A058446Source2236081408416666 2 =
5000060065066660656065066555556
0 5 7---combination impossible
0 5 8---combination impossible
0 5 9A058447A058448Source771395165003 2 =
595050500590005595990009
0 6 7A058449A058450Source26012881552428213576 2 =
676670006660660066767076066770670707776
0 6 8---combination impossible
0 6 9A058451A058452Source30000101109940614 2 =
900006066606660060090966606696996
0 7 8---combination impossible
0 7 9A058453A058454Source8819172285373497 2 =
77777799799099990007000790009009
0 8 9A058455A058456Source301345331969667 2 =
90809009099908808089808090889
1 2 3A030175A030174Source557963558954625926861 2 =
311323333121312322332133323111223321313321
1 2 4A053880A053881SourceOne infinite pattern
(3n)8 2 = (1n)4(2n-1)44  [n>=1]
379766258564954821662 2 =
144222411144424121442444112111142224442244
1 2 5A031153A031153SourceThree 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]
1102340925268369741032335 2 =
1215155515521525522215211555521512552151515552225
1 2 6A053882A053883Source47130268582155593596 2 =
2221262216626122626662262611111116211216
1 2 7A053884A053885Source130834904430015239 2 =
17117772217211221211117217772227121
1 2 8A053886A053887SourceOne infinite pattern
(3n)59 2 = (1n)28(2n-1)881  [n>=1]
34377642169166984891 2 =
1181822281111288118221882821111822281881
1 2 9A053888A053889Source459556524411439511 2 =
211192199129121999192121999211919121
1 3 4A053890A053891Source21079405433537116521 2 =
444341333431434111311131411343131143441
1 3 5---impossible since all 3 digits odd
1 3 6A053892A053893Source18266544874814631 2 =
333666661663616663311166611666161
1 3 7---impossible since all 3 digits odd
1 3 8A053894A053895Source286074095527510693610891 2 =
81838388131883313833383381811133318888113813881
1 3 9---impossible since all 3 digits odd
1 4 5A053896A053897Source735604795060122 =
5411144145154411455544144144
1 4 6A027677A027676Source 1
Source 2		10789398111648380852704 2 =
116411111611641646616166611414441166144111616
1 4 7A053898A053899Source2177492084289725902412 2 =
4741471777144414774147714111744447747417744
1 4 8A053900A053901Source1346900557360669225841779 2 =
1814141111418481411488184148411114184811141884841
1 4 9A027675A006716 *Source 1
Source 2
Source 3		(* seq by njas)
648070211589107021 2 =
419994999149149944149149944191494441
1 5 6A053902A053903SourceOne infinite pattern
(3n)4 2 = (1n+1)(5n)6  [n>=0]
74240565428619726479296 2 =
5511661555161166511651661611666516615516655616
1 5 7---impossible since all 3 digits odd
1 5 8A053904A053905Source2412237158970509643109 2 =
5818888111118115811551585855811558551185881
1 5 9---impossible since all 3 digits odd
1 6 7A053906A053907Source1292931424 2 =
1671671667166667776
1 6 8A053908A053909Source1080794204132598414568541 2 =
1168116111686616811868118886618818666111186868681
1 6 9A053910A053911Source34149670012924966713187 2 =
1166199961991666696199696161116991961919696969
1 7 8A053912A053913Source279067891 2 =
77878887787187881
1 7 9---impossible since all 3 digits odd
1 8 9A053914A053915Source2969848344609859 2 =
8819999189981919818818919999881
2 3 4A053916A053917Source205483392086668 2 =
42223424423443333243223342224
2 3 5A053918A053919Source159789024443333515 2 =
25532532332552235533223325522255225
2 3 6A058457A058458Source2514602599284156 2 =
6323226232326633633323632632336
2 3 7---combination impossible
2 3 8---combination impossible
2 3 9A053920A053921Source14940646884386874573 2 =
223222929323939222223332232999233932329
2 4 5A031154A031152SourceThree 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]
473545618135383472428338 2 =
224245452455222424254455242452522555442545442244
2 4 6A053922A053923SourceOne infinite pattern
(6n)8 2 = (4n)6(2n)4  [n>=0]
1562826497869300353470568 2 =
2442426662442422262266222264624646466242442242624
2 4 7A058459A058460Source47146022358675418 2 =
2222747424244722424422447477474724
2 4 8A027679A027678Source 1
Source 2
Source 3		2069416058768323727702022 2 =
4282482824288222284288288882288482848444822888484
2 4 9A053924A053925Source547259530974381470838 2 =
299492994242299992492444422992424244422244
2 5 6A030486A030484Source81401637345465395512991484 2 =
6626226562522666562566262626266252566552622656522256
2 5 7A030487A030485SourceOne infinite pattern
1(6n)5 2 = 2(7n)(2n+1)5  [n>=0]
870185357137045415 2 =
757222555775727275772757755772522225
2 5 8A053926A053927Source159013392166264585 2 =
25285258888222255288288552225222225
2 5 9A053928A053929Source1598601020441309256565 2 =
2555525222555995255555255252529952995599225
2 6 7A058461A058462Source150573163864701424 2 =
22672277676226226672276776667627776
2 6 8---combination impossible
2 6 9A053930A053931Source 1
Source 2
Source 3		47567102808870567435673 2 =
2262629269629662226292666922999622262992962929
2 7 8---combination impossible
2 7 9A053932A053933Source14907304327 2 =
222227722297792922929
2 8 9A053934A053935Source5405829167667 2 =
29222988989999289998222889
3 4 5A053936A053937Source21343231796858797962 2 =
455533543534444433554534333443535353444
3 4 6A053938A053939Source6666680833328031344 2 =
44444633333463334436443663666646446336
3 4 7A058463A058464Source1864878916830083039312 2 =
3477773374437343773773333743737447337433344
3 4 8A053940A053941Source5906300402396058566810062 2 =
34884384443343843348888488484833488344838384443844
3 4 9A053942A053943Source185605616817891584607 2 =
34449444994349999433343349994949439344449
3 5 6A053944A053945Source23527926717739784 2 =
553563335635333565566365536366656
3 5 7---impossible since all 3 digits odd
3 5 8---combination impossible
3 5 9---impossible since all 3 digits odd
3 6 7A053946A053947Source1932967502917049474 2 =
3736363367333373666777367633763676676
3 6 8A058465A058466Source183539278812156 2 =
33686666866886336386333368336
3 6 9A053948A053949Source18430047920827535573187 2 =
339666666363999366939366969366969936633336969
3 7 8---combination impossible
3 7 9---impossible since all 3 digits odd
3 8 9A058467A058468Source199974958167 2 =
39989983893893399999889
4 5 6A030177A030176Source675754811056988742949784 2 =
456644564666666555445565455644644555565545646656
4 5 7A053950A053951Source8629863583949388 2 =
74474545477575775744575745574544
4 5 8A053952A053953Source767175898056538 2 =
588558858558855585845444545444
4 5 9A053954A053955Source703957001491895099962643 2 =
495555459949459999994555545994544549499995545449
4 6 7A053956A053957Source815277035409723028858892 2 =
664676644466466777446466766667744446667647467664
4 6 8A053958A053959Source26204321529981784378 2 =
686666466846666884868486448488884846884
4 6 9A053960A053961Source9718263579193026119075264 2 =
94444646994669646646646969664664969996646496669696
4 7 8A053962A053963Source22019193553462506122 2 =
484844884744844787448744774844887478884
4 7 9A053964A053965Source8819171038 2 =
77777777797497997444
4 8 9A053966A053967SourceOne infinite pattern
(6n)7 2 = (4n+1)(8n)9  [n>=0]
6670081667 2 =
44489989444449498889
5 6 7A053968A053969Source2562353735836753390526 2 =
6565656667556566576676575665776756666556676
5 6 8  SourceOnly this one rather trivial solution is known
816 2 =
665856
5 6 9A053970A053971Source834023722663550236 2 =
695595569965566559565695699695655696
5 7 8---combination impossible
5 7 9---impossible since all 3 digits odd
5 8 9A058469A058470Source92496431583 2 =
8555589855588599885889
6 7 8  Source( No solutions under 10 50
info from Hisanori Mishima's website )
6 7 9A053972A053973Source9831977256725526 2 =
96667776776767999797799699976676
6 8 9A053974A053975Source2588184048685235767383 2 =
6698696669868698868988986669668968886668689
7 8 9A058471A058472Source9949370777987917 2 =
98989978877879888789778997998889

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[© 1996,2006] - Last modified : January 13, 2007.
Patrick De Geest - Belgium flag - Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com