A220698 -id:A220698 - cp's OEIS frontend

A220752 Terms of A220698 that appear in A224218.

3854, 3854, 3035, 3035, 3035, 3035, 3854, 4644, 4644, 4644, 4644, 4644, 3854, 15846, 4644, 4644, 4644, 4644, 4644, 22918, 15846, 15846, 10225, 10225, 10225, 10225, 15846, 22918, 15846, 13364, 13364, 13364, 13364, 10225, 10225, 10225, 10225, 15846, 13364, 13364, 22918, 45012

Offset: 1

Alex Ratushnyak, Apr 13 2013

Terms of A220698 excluding terms that do not appear in A224218.
Indices of XOR-positive triangular numbers such that the generated triangular number is also XOR-positive (definition: triangular(i) is XOR-positive if triangular(i) XOR triangular(i+1) = triangular(k) for some k). XOR is the bitwise logical exclusive-or operator.
Conjecture: the sequence is infinite.
The subsequence with only odd terms begins: 3035, 3035, 3035, 3035, 10225, 10225, 10225, 10225, 10225, 10225, 10225, 10225, 171449, 171449, 236985, 171449, 339249.

Cf. A000217, A224218, A220698.

#include 
typedef unsigned long long U64;
U64 rootTriangular(U64 a) {
    U64 sr = 1L<<32, s, b;
    if (a < (sr/2)*(sr+1)) {
          sr>>=1;
          while (a < sr*(sr+1)/2)  sr>>=1;
    }
    for (b = sr>>1; b; b>>=1) {
            s = sr+b;
            if (a >= s*(s+1)/2)  sr = s;
    }
    return sr;
}
int main() {
  U64 a, n, r, t;
  for (n=0; n < (1L<<32)-1; n++) {
    a = (n*(n+1)/2) ^ ((n+1)*(n+2)/2);
    t = rootTriangular(a);
    if (a == t*(t+1)/2) {
        a ^= (t+1)*(t+2)/2;
        r = rootTriangular(a);
        if (a == r*(r+1)/2)  printf("%llu, ", t);
    }
  }
}

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.

Original entry on oeis.org

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

Alex Ratushnyak, Apr 01 2013

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.

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

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

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

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

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 *)

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=',')

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A220752 Terms of A220698 that appear in A224218.

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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.

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