A335132 Numbers whose binary expansion generates 3-fold rotationally symmetric EQ-triangles.

0, 1, 3, 5, 7, 9, 15, 17, 31, 33, 63, 65, 73, 119, 127, 129, 255, 257, 297, 349, 373, 395, 419, 471, 511, 513, 585, 653, 709, 827, 883, 951, 1023, 1025, 1193, 1879, 2047, 2049, 2145, 2225, 2257, 3887, 3919, 3999, 4095, 4097, 4321, 4681, 4777, 5501, 5533, 5941

Offset: 1

Author

Rémy Sigrist, May 24 2020

Keywords

• nonn
• base

Comments

For any nonnegative number n, the EQ-triangle for n is built by taking as first row the binary expansion of n (without leading zeros), having each entry in the subsequent rows be the EQ of the two values above it (a "1" indicates that these two values are equal, a "0" indicates that these values are different).

The second row in such a triangle has binary expansion given by A279645.

If m belongs to this sequence, then A030101(m) also belongs to this sequence.

All positive terms are odd.

This sequence is a variant of A334556; here we use bitwise EQ, there bitwise XOR.

Examples

			For 349:
- the binary expansion of 349 is "101011101",
- the corresponding EQ-triangle is (with dots instead of 0's for clarity):
     1 . 1 . 1 1 1 . 1
      . . . . 1 1 . .
       1 1 1 . 1 . 1
        1 1 . . . .
         1 . 1 1 1
          . . 1 1
           1 . 1
            . .
             1
- this triangle has 3-fold rotational symmetry, so 349 belongs to this sequence.
		

Links

• Rémy Sigrist, Illustration of initial terms
• Wikipedia, Logical equality
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A279645, A334556.

Programs

PARI

is(n) = {
    my (b=binary(n), p=b);
    for (k=1, #b,
        if (b[k]!=p[#p], return (0));
        if (p[1]!=b[#b+1-k], return (0));
        p = vector(#p-1, k, p[k]==p[k+1]);
    );
    return (1);
}