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A220689 Triangular numbers generated in A224218. That is, the triangular numbers generated by the operation triangular(i) XOR triangular(i+1) along increasing i.

%I A220689 #33 Aug 07 2023 07:48:21
%S A220689 1,21,21,105,105,105,105,946,946,666,1653,666,1378,946,1225,946,4005,
%T A220689 1378,4005,1378,7381,1225,1378,1653,2485,4005,31125,4005,4005,4005,
%U A220689 2485,13861,13861,5356,4005,7381,5356,5356,7381,4005,5356,29161,12561,12561,4186,4186,4186,4186
%N A220689 Triangular numbers generated in A224218. That is, the triangular numbers generated by the operation triangular(i) XOR triangular(i+1) along increasing i.
%F A220689 a(n) = A000217(A224218(n)) XOR A000217(A224218(n)+1).
%p A220689 read("transforms") ;
%p A220689 A220689 := proc(n)
%p A220689     i := A224218(n) ;
%p A220689     XORnos(A000217(i),A000217(i+1)) ;
%p A220689 end proc: # _R. J. Mathar_, Apr 23 2013
%t A220689 nmax = 100;
%t A220689 pmax = 2 nmax^2; (* increase coeff 2 if A224218 is too short *)
%t A220689 A224218 = Join[{0}, Flatten[Position[Partition[Accumulate[Range[pmax]], 2, 1], _?(OddQ[Sqrt[1 + 8 BitXor[#[[1]], #[[2]]]]]&), {1}, Heads -> False]]];
%t A220689 a[n_] := Module[{i}, i = A224218[[n]]; BitXor[PolygonalNumber[i], PolygonalNumber[i+1]]];
%t A220689 Table[a[n], {n, 1, nmax}] (* _Jean-François Alcover_, Aug 07 2023, after _Harvey P. Dale_ in A224218 *)
%o A220689 (Python)
%o A220689 def rootTriangular(a):
%o A220689     sr = 1<<33
%o A220689     while a < sr*(sr+1)//2:
%o A220689       sr>>=1
%o A220689     b = sr>>1
%o A220689     while b:
%o A220689         s = sr+b
%o A220689         if a >= s*(s+1)//2:
%o A220689           sr = s
%o A220689         b>>=1
%o A220689     return sr
%o A220689 for i in range(1<<12):
%o A220689         s = (i*(i+1)//2) ^ ((i+1)*(i+2)//2)
%o A220689         t = rootTriangular(s)
%o A220689         if s == t*(t+1)//2:
%o A220689             print(str(s), end=',')
%Y A220689 Cf. A000217, A224218, A224511.
%K A220689 nonn
%O A220689 1,2
%A A220689 _Alex Ratushnyak_, Apr 13 2013