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

1, 21, 21, 105, 105, 105, 105, 946, 946, 666, 1653, 666, 1378, 946, 1225, 946, 4005, 1378, 4005, 1378, 7381, 1225, 1378, 1653, 2485, 4005, 31125, 4005, 4005, 4005, 2485, 13861, 13861, 5356, 4005, 7381, 5356, 5356, 7381, 4005, 5356, 29161, 12561, 12561, 4186, 4186, 4186, 4186

Offset: 1

Author

Alex Ratushnyak, Apr 13 2013

Keywords

• nonn

Crossrefs

Cf. A000217, A224218, A224511.

Programs

Maple

read("transforms") ;
A220689 := proc(n)
    i := A224218(n) ;
    XORnos(A000217(i),A000217(i+1)) ;
end proc: # R. J. Mathar, Apr 23 2013

Mathematica

nmax = 100;
pmax = 2 nmax^2; (* increase coeff 2 if A224218 is too short *)
A224218 = Join[{0}, Flatten[Position[Partition[Accumulate[Range[pmax]], 2, 1], _?(OddQ[Sqrt[1 + 8 BitXor[#[[1]], #[[2]]]]]&), {1}, Heads -> False]]];
a[n_] := Module[{i}, i = A224218[[n]]; BitXor[PolygonalNumber[i], PolygonalNumber[i+1]]];
Table[a[n], {n, 1, nmax}] (* Jean-François Alcover, Aug 07 2023, after Harvey P. Dale in A224218 *)

Python

def rootTriangular(a):
    sr = 1<<33
    while a < sr*(sr+1)//2:
      sr>>=1
    b = sr>>1
    while b:
        s = sr+b
        if a >= s*(s+1)//2:
          sr = s
        b>>=1
    return sr
for i in range(1<<12):
        s = (i*(i+1)//2) ^ ((i+1)*(i+2)//2)
        t = rootTriangular(s)
        if s == t*(t+1)//2:
            print(str(s), end=',')

Formula

a(n) = A000217(A224218(n)) XOR A000217(A224218(n)+1).