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

Author

Jack Zhang, Apr 28 2022

Keywords

• nonn
• easy

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)

Crossrefs

Cf. A001969, A048725.

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

Programs

PARI

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

Extensions

More terms from David A. Corneth, Apr 28 2022