%I #45 Apr 09 2022 06:36:48
%S 0,1,3,6,10,55,66,300,666,990,3003,5050,10011,66066,500500,600060,
%T 50005000,5000050000,500000500000,50000005000000,5000000050000000,
%U 500000000500000000,50000000005000000000,5000000000050000000000,500000000000500000000000,50000000000005000000000000
%N Triangular numbers whose nonzero digits are all the same.
%C This sequence may correspond to "monochromatic step squads" in the British animation "Numberblocks".
%C Conjecture: the largest term in this sequence whose nonzero digits are not 5 is 600060.
%t (* Method1 *)
%t NonZeroQ[n_Integer] := n != 0; Select[
%t Table[n (n + 1)/2, {n, 0, 1000000}],
%t Length[Tally[Select[IntegerDigits[#], NonZeroQ]]] == 1 &]
%t (* Method2 *)
%t Sort[Select[
%t Flatten[Outer[Times,
%t Table[FromDigits[IntegerDigits[n, 2]], {n, 2^16 - 1}], Range[9]]],
%t IntegerQ[Sqrt[8 # + 1]] &]]
%o (Python)
%o from sympy import integer_nthroot
%o from sympy.utilities.iterables import multiset_permutations
%o def istri(n): return integer_nthroot(8*n+1, 2)[1]
%o def zplus1(digits):
%o if digits == 1: yield 0
%o for d1 in "123456789":
%o digset = "0"*(digits-1) + d1*(digits-1)
%o for mp in multiset_permutations(digset, digits-1):
%o t = int(d1 + "".join(mp))
%o yield t
%o def afind(maxdigits):
%o for digits in range(1, maxdigits+1):
%o for t in zplus1(digits):
%o if istri(t):
%o print(t, end=", ")
%o afind(22) # _Michael S. Branicky_, Mar 02 2022
%o (PARI) isok(k) = my(d=digits(k*(k+1)/2)); d = select(x->(x!=0), d); #Set(d)<=1;
%o lista(nn) = {for (n=0, nn, if (isok(n), print1(n*(n+1)/2, ", ")););} \\ _Michel Marcus_, Mar 02 2022
%Y Supersequence of A037156.
%Y Cf. A000217, A118668, A125289, A343811.
%Y Cf. A352148 (indices of these triangular numbers).
%K nonn,base
%O 1,3
%A _Steven Lu_, Mar 02 2022
%E a(24)-a(25) from _Michael S. Branicky_, Mar 02 2022