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

1,2

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.

Links

Table of n, a(n) for n=1..63.

Formula

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

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 *)

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(t), end=', ')

Crossrefs

Cf. A000217, A224218, A220689.

Sequence in context: A229828 A141378 A003871 * A315802 A315803 A315804

Adjacent sequences: A220695 A220696 A220697 * A220699 A220700 A220701

Keyword

nonn

Author

Alex Ratushnyak, Apr 13 2013

Status

approved