%I #32 Jan 27 2023 12:23:43

%S 0,1,3,6,55,66,666

%N Triangular numbers with all digits the same.

%C Escott (1905) proved that there are no more terms with fewer than 30 digits. The complete proof that there are no more terms was given by Ballew and Weger (1972). - _Amiram Eldar_, Jan 22 2022

%D L. E. Dickson, History of the Theory of Numbers, Vol. II, p. 33, Chelsea NY, 1952.

%D E. B. Escott, Math. Quest. Educational Times, New Series, Vol. 8 (1905), pp. 33-34. - _N. J. A. Sloane_, Mar 31 2014

%H David W. Ballew and Ronald C. Weger, <a href="https://sdaos.org/wp-content/uploads/pdfs/Vol%2051%201972/72p52.pdf">Triangular Numbers with Repeated Digits</a>, Proc. S. D. Acad. Sci., Vol. 51 (1972), pp. 52-55.

%H David W. Ballew and Ronald C. Weger, <a href="https://yutaka-nishiyama.sakura.ne.jp/math/repdigit.pdf">Repdigit triangular numbers</a>, J. Rec. Math., Vol. 8, No. 2 (1975-76), pp. 96-98.

%H Bir Kafle, Florian Luca and Alain Togbé, <a href="https://www.fq.math.ca/Abstracts/56-4/kafle.pdf">Triangular Repblocks</a>, Fibonacci Quart., Vol. 56, No. 4 (2018), pp. 325-328.

%H C. E. Youngman, <a href="https://archive.org/details/educationaltimes58educ/page/87/mode/1up">Problem 15648</a>, Educational Times, Vol. 58, 1905, p. 87; with a solution by E. B. Escott.

%F A118668(a(n)) = 1. - _Reinhard Zumkeller_, Jul 11 2015

%t Select[Union[Flatten[Table[FromDigits[PadRight[{},n,k]],{n,3},{k,0,9}]]],OddQ[ Sqrt[8#+1]]&] (* _Harvey P. Dale_, Feb 11 2020 *)

%Y Cf. A213516 (triangular numbers having only 1 or 2 different digits).

%Y Cf. A118668.

%K fini,full,nonn,base

%O 1,3

%A _Felice Russo_

%E 0 inserted by _T. D. Noe_, Jun 22 2012