A220698 Indices of triangular numbers generated in A224218.

1, 6, 6, 14, 14, 14, 14, 43, 43, 36, 57, 36, 52, 43, 49, 43, 89, 52, 89, 52, 121, 49, 52, 57, 70, 89, 249, 89, 89, 89, 70, 166, 166, 103, 89, 121, 103, 103, 121, 89, 103, 241, 158, 158, 91, 91, 91, 91, 241, 166, 166, 103, 121, 103, 103, 121, 103, 121, 225, 225, 497, 216, 334

Offset: 1

Author

Alex Ratushnyak, Apr 13 2013

Keywords

• nonn

Comments

Indices of triangular numbers in A220689. That is, S = triangular(i) XOR triangular(i+1); increment i; if S is a triangular number then index of S is appended to a(n). Initially i=0. XOR is the binary logical exclusive-or operator.

Crossrefs

Cf. A000217, A224218, A220689.

Programs

Maple

A220698 := proc(n)
    A127648(A220689(n)-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, t}, i = A224218[[n]]; t = BitXor[PolygonalNumber[i], PolygonalNumber[i + 1]]; (Sqrt[8 t + 1] - 1)/2];
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(t), end=',')

Formula

a(n) = i where A000217(i) = A220689(n).