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%I #35 May 03 2023 23:30:18
%S 1,10,21,120,210,2211,10011,20100,112101,222111,2001000,22221111,
%T 110120220,122000010,200010000,1210000221,2222211111,12001110201,
%U 20000100000,122021211021,222222111111,2000001000000,12201101000011,22222221111111,200000010000000
%N Triangular numbers composed of digits {0,1,2}.
%C Cross-references to similar sequences:
%C 012 A119033 013 A119035 014 A119037 015 A119039 016 A119041
%C 017 A119043 018 A119045 019 A119047 023 A119049 024 A218390
%C 025 A119051 026 A119053 027 A218397 028 A119055 029 A119057
%C 034 A119059 035 A119061 036 A119063 037 A119065 038 A119067
%C 039 A119069 045 A119071 046 A119073 047 A218399 048 A119075
%C 049 A119077 056 A119079 057 A119081 058 A119083 059 A119085
%C 067 A119087 068 A119089 069 A119091 078 A119093 079 A218401
%C 089 A119095 123 A119097 124 A119099 125 A119101 126 A119103
%C 127 A119105 128 A119107 129 A119109 134 A119111 135 A119113
%C 136 A119115 137 A119117 138 A119119 139 A119121 145 A119123
%C 146 A119125 147 A079654 148 A119128 149 A119130 156 A119132
%C 157 A119134 158 A119136 159 A119138 167 A119140 168 A119142
%C 169 A119144 178 A119146 179 A119148 189 A119150 234 {3}
%C 235 A119152 236 A119154 237 {3} 238 A119156 239 {3}
%C 245 A119158 246 A119160 247 { } 248 A119162 249 { }
%C 256 A119164 257 A119166 258 A119168 259 A119170 267 A119172
%C 268 A119174 269 A119176 278 A119178 279 { } 289 A119180
%C 345 A119182 346 A119184 347 {3} 348 {3} 349 {3}
%C 356 A119186 357 A119188 358 A119190 359 A119192 367 A119194
%C 368 A119196 369 A119198 378 A119200 379 {3} 389 {3}
%C 456 A119202 457 A119204 458 A119206 459 A119208 467 A119210
%C 468 A119212 469 A119214 478 A119216 479 { } 489 { }
%C 567 A119218 568 A119220 569 A119222 578 A119224 579 A119226
%C 589 A119228 678 A119230 679 A119232 689 A119234 789 A119236
%C Entries marked "{ }" correspond to empty sequences: for every triangular number t, the residue t mod 100 contains at least one digit other than the three specified digits.
%C Entries marked "{3}" correspond to sequences containing only the single term 3: for every triangular number t != 3, the residue t mod 100 contains at least one digit other than the three specified digits.
%C (Proof: No triangular number ends in 2, 4, 7, or 9; every triangular number ending in 8 ends in 28 or 78; every triangular number ending in 3, other than the single-digit triangular number 3, ends in 03 or 53.) [Edited by _Jon E. Schoenfield_, May 02 2023]
%C Note that the first 36 sequences that are listed above do not contain "0" as the first term although 0 is a triangular number. In other words, sequences focus on the positive triangular numbers. - _Altug Alkan_, May 02 2016
%C a(n) == 1 or a(n) == 0 (mod 10). - _Chai Wah Wu_, Nov 30 2018
%H Giovanni Resta, <a href="http://www.numbersaplenty.com/tr/tr012.html">Tridigital Triangular Numbers</a>.
%F a(n) = A000217(A119034(n)). - _Tyler Busby_, Mar 31 2023
%t Rest[Select[FromDigits/@Tuples[{0, 1, 2}, 10], IntegerQ[(Sqrt[8 # + 1] - 1)/2] &]] (* _Vincenzo Librandi_, Dec 18 2015 *)
%o (Magma) [t: n in [1..2*10^7] | Set(Intseq(t)) subset {0,1,2} where t is n*(n+1) div 2]; // _Vincenzo Librandi_, Dec 18 2015
%o (PARI) isok(n) = ispolygonal(n, 3) && (vecmax(digits(n)) <= 2); \\ _Michel Marcus_, Dec 18 2015
%Y Cf. A000217, A058412, A119034.
%Y Cf. A213516 (triangular numbers having only two different digits).
%K nonn,base
%O 1,2
%A _Giovanni Resta_, May 10 2006
%E a(24)-a(25) from _Vincenzo Librandi_, Dec 18 2015
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