A193232 -id:A193232 - cp's OEIS frontend

A220518 Numbers n such that A193232(n) is a triangular number.

0, 1, 5, 9, 45, 49, 101, 153, 1569, 7163, 7171, 8162, 18974, 18976, 24467, 33490, 60290, 63046, 359539, 551494, 1769418, 2813691, 4140392, 4649729, 6675935, 9653486, 59131393, 158169499, 243345386, 588183781, 1315697727, 1387290631, 1522472645, 1879098546

Offset: 1

Alex Ratushnyak, Mar 27 2013

Numbers k such that bitwise XOR of first k triangular numbers is a triangular number.

Cf. A193232.

Cf. A145827 (bitwise XOR of first k squares is a square).

#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=0, i, t;
  for (i=0; i < 1L<<32; ++i) {
      a ^= i*(i+1)/2;
      t = rootTriangular(a);
      if (a == t*(t+1)/2)  printf("%llu\n", i);
  }
  return 0;
}

A145768 a(n) = the bitwise XOR of squares of first n natural numbers.

Original entry on oeis.org

0, 1, 5, 12, 28, 5, 33, 16, 80, 1, 101, 28, 140, 37, 225, 0, 256, 33, 357, 12, 412, 37, 449, 976, 400, 993, 325, 924, 140, 965, 65, 896, 1920, 961, 1861, 908, 1692, 965, 1633, 912, 1488, 833, 1445, 668, 1292, 741, 2721, 512, 2816, 609, 2981, 396, 2844, 485

Offset: 0

Vladimir Reshetnikov, Oct 18 2008

Up to n=10^8, a(15) is the only zero term and a(1)=a(9) are the only terms for which a(n)=1. Can it be proved that any number can only appear a finite number of times in this sequence? [M. F. Hasler, Oct 20 2008]
Even terms occur at A014601, odd terms at A042963; A010873(a(n))=A021913(n+1). - Reinhard Zumkeller, Jun 05 2012
If squares occur, they must be at indexes != 2 or 5 (mod 8). - Roderick MacPhee, Jul 17 2017

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
StackExchange, Perfect squares in a XOR-Sum of perfect squares

Cf. A003815, A145827, A145828, A145829, A145830, A145831. [M. F. Hasler, Oct 20 2008]
Cf. A193232.
Cf. A000290.

import Data.Bits (xor)
a145768 n = a145768_list !! n
a145768_list = scanl1 xor a000290_list  -- Reinhard Zumkeller, Jun 05 2012

A[0]:= 0:
for n from 1 to 100 do A[n]:= Bits:-Xor(A[n-1],n^2) od:
seq(A[i],i=0..100); # Robert Israel, Dec 08 2019

Rest@ FoldList[BitXor, 0, Array[#^2 &, 50]]

an=0; for( i=1,50, print1(an=bitxor(an,i^2),",")) \\ M. F. Hasler, Oct 20 2008

al(n)=local(m);vector(n,k,m=bitxor(m,k^2))

from functools import reduce
from operator import xor
def A145768(n):
    return reduce(xor, [x**2 for x in range(n+1)]) # Chai Wah Wu, Aug 08 2014

a(n)=1^2 xor 2^2 xor ... xor n^2.

cp's OEIS Frontend

A220518 Numbers n such that A193232(n) is a triangular number.

Views

Author

Keywords

Comments

Crossrefs

Programs

C

A145768 a(n) = the bitwise XOR of squares of first n natural numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

Python

Formula