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

1,2

Links

Table of n, a(n) for n=1..48.

Formula

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

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 *)

Prog

(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=', ')

Crossrefs

Cf. A000217, A224218, A224511.

Sequence in context: A371100 A048245 A246034 * A056485 A056475 A219913

Adjacent sequences: A220686 A220687 A220688 * A220690 A220691 A220692

Keyword

nonn

Author

Alex Ratushnyak, Apr 13 2013

Status

approved