A119033 Triangular numbers composed of digits {0,1,2}.

1, 10, 21, 120, 210, 2211, 10011, 20100, 112101, 222111, 2001000, 22221111, 110120220, 122000010, 200010000, 1210000221, 2222211111, 12001110201, 20000100000, 122021211021, 222222111111, 2000001000000, 12201101000011, 22222221111111, 200000010000000

Offset

1,2

Comments

Cross-references to similar sequences:

012 A119033 013 A119035 014 A119037 015 A119039 016 A119041

017 A119043 018 A119045 019 A119047 023 A119049 024 A218390

025 A119051 026 A119053 027 A218397 028 A119055 029 A119057

034 A119059 035 A119061 036 A119063 037 A119065 038 A119067

039 A119069 045 A119071 046 A119073 047 A218399 048 A119075

049 A119077 056 A119079 057 A119081 058 A119083 059 A119085

067 A119087 068 A119089 069 A119091 078 A119093 079 A218401

089 A119095 123 A119097 124 A119099 125 A119101 126 A119103

127 A119105 128 A119107 129 A119109 134 A119111 135 A119113

136 A119115 137 A119117 138 A119119 139 A119121 145 A119123

146 A119125 147 A079654 148 A119128 149 A119130 156 A119132

157 A119134 158 A119136 159 A119138 167 A119140 168 A119142

169 A119144 178 A119146 179 A119148 189 A119150 234 {3}

235 A119152 236 A119154 237 {3} 238 A119156 239 {3}

245 A119158 246 A119160 247 { } 248 A119162 249 { }

256 A119164 257 A119166 258 A119168 259 A119170 267 A119172

268 A119174 269 A119176 278 A119178 279 { } 289 A119180

345 A119182 346 A119184 347 {3} 348 {3} 349 {3}

356 A119186 357 A119188 358 A119190 359 A119192 367 A119194

368 A119196 369 A119198 378 A119200 379 {3} 389 {3}

456 A119202 457 A119204 458 A119206 459 A119208 467 A119210

468 A119212 469 A119214 478 A119216 479 { } 489 { }

567 A119218 568 A119220 569 A119222 578 A119224 579 A119226

589 A119228 678 A119230 679 A119232 689 A119234 789 A119236

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.

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.

(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]

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

a(n) == 1 or a(n) == 0 (mod 10). - Chai Wah Wu, Nov 30 2018

Links

Table of n, a(n) for n=1..25.

Giovanni Resta, Tridigital Triangular Numbers.

Formula

a(n) = A000217(A119034(n)). - Tyler Busby, Mar 31 2023

Mathematica

Rest[Select[FromDigits/@Tuples[{0, 1, 2}, 10], IntegerQ[(Sqrt[8 # + 1] - 1)/2] &]] (* Vincenzo Librandi, Dec 18 2015 *)

Prog

(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
(PARI) isok(n) = ispolygonal(n, 3) && (vecmax(digits(n)) <= 2); \\ Michel Marcus, Dec 18 2015

Crossrefs

Cf. A000217, A058412, A119034.

Cf. A213516 (triangular numbers having only two different digits).

Sequence in context: A306208 A001739 A072805 * A171067 A121807 A133163

Adjacent sequences: A119030 A119031 A119032 * A119034 A119035 A119036

Keyword

nonn,base

Author

Giovanni Resta, May 10 2006

Extensions

a(24)-a(25) from Vincenzo Librandi, Dec 18 2015

Status

approved