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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 (list; graph; refs; listen; history; text; internal format)
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: A284342 A198470 A245379 * A076485 A325387 A272860

Adjacent sequences: A224215 A224216 A224217 * A224219 A224220 A224221

KEYWORD

nonn,base

AUTHOR

Alex Ratushnyak, Apr 01 2013

STATUS

approved

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Last modified January 27 12:24 EST 2023. Contains 359840 sequences. (Running on oeis4.)