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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.
5
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
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
KEYWORD
nonn,base
AUTHOR
Alex Ratushnyak, Apr 01 2013
STATUS
approved