A353167 Polynomials over GF(2) that are divisible by (x+1)^2 = x^2+1, encoded as binary numbers.

0, 5, 10, 15, 17, 20, 27, 30, 34, 39, 40, 45, 51, 54, 57, 60, 65, 68, 75, 78, 80, 85, 90, 95, 99, 102, 105, 108, 114, 119, 120, 125, 130, 135, 136, 141, 147, 150, 153, 156, 160, 165, 170, 175, 177, 180, 187, 190, 195, 198, 201, 204, 210, 215, 216, 221, 225, 228, 235

Offset

1,2

Comments

Terms of A048725, sorted.

See also A001969 for those divisible by x+1 (and obviously the present sequence is a subsequence of that one).

From Kevin Ryde, Jul 22 2022: (Start)

Integers with an even number of 1-bits at even positions, and an even number of 1-bits at odd positions, and so all k with A355487(k) = 0.

Among four integers 4*i ..4*i+3, exactly one is a term here so that a(n) can be calculated by appending two bits to n-1 to ensure the two 1-bit counts are even, so a(n) = 4*(n-1) + A355487(n-1).

(End)

Links

Table of n, a(n) for n=1..59.

Prog

(PARI) a(n) = n--; n<<2 + if(n, fold(bitxor, digits(n, 4))); \\ Kevin Ryde, Jul 01 2022

Crossrefs

Cf. A001969, A048725.

Cf. A355487 (mod 4), A341389 (mod 2).

Sequence in context: A236844 A271468 A313669 * A254063 A313670 A321513

Adjacent sequences: A353164 A353165 A353166 * A353168 A353169 A353170

Keyword

nonn,easy

Author

Jack Zhang, Apr 28 2022

Extensions

More terms from David A. Corneth, Apr 28 2022

Status

approved