A224218 Indices of XOR-positive triangular numbers. That is, numbers n such that triangular(n) XOR triangular(n+1) is a triangular number, where XOR is the bitwise logical XOR operator.

0, 12, 18, 24, 40, 86, 102, 177, 333, 357, 628, 665, 669, 689, 840, 845, 860, 861, 1124, 1185, 1196, 1206, 1377, 1418, 1706, 1890, 1906, 1956, 2138, 2204, 2388, 2524, 2588, 2843, 2970, 2994, 3035, 3107, 3154, 3234, 3299, 3606, 3824, 3854, 4005, 4021, 4169, 4185, 4568, 4580

Offset

1,2

Comments

Generated triangular numbers: A220689(n) = A000217(a(n)) XOR A000217(a(n)+1).

Indices of triangular numbers in A220689: A220698.

Terms of A220698 that appear in this sequence: A220752.

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Example

Triangular(18) XOR triangular(19) = 171 XOR 190 = 21, because 21 is a triangular number, 18 is in the sequence.

Maple

read("transforms") ;
isA000217 := proc(n)
    local t1;
    t1:=floor(sqrt(2*n));
    if n = t1*(t1+1)/2 then
        return true
    else
        return false;
    end if;
end proc:
isA224218 := proc(n)
    XORnos(A000217(n), A000217(n+1)) ;
    isA000217(%) ;
end proc:
A224218 := proc(n)
    option remember;
    if n = 1 then
        0;
    else
        for a from procname(n-1)+1 do
            if isA224218(a) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Apr 23 2013

Mathematica

Join[{0}, Flatten[Position[Partition[Accumulate[Range[5000]], 2, 1], _?(OddQ[ Sqrt[1+8BitXor[#[[1]], #[[2]]]]]&), {1}, Heads->False]]] (* Harvey P. Dale, Dec 05 2014 *)

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(i), end=', ')

Crossrefs

Cf. A000217, A220689, A220698, A220752, A221643, A220518.

Sequence in context: A198470 A245379 A363814 * A076485 A325387 A272860

Adjacent sequences: A224215 A224216 A224217 * A224219 A224220 A224221

Keyword

nonn,base

Author

Alex Ratushnyak, Apr 01 2013

Status

approved