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

Author

Alex Ratushnyak, Apr 01 2013

Keywords

• nonn
• base

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.

Examples

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

Links

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

Crossrefs

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

Programs

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 *)

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=',')