A132454 First primitive GF(2)[X] polynomials of degree n and minimal number of terms, expressed as -k for X^n+X^k+1, else with X^n suppressed.
1, -1, -1, -1, -2, -1, -1, 29, -4, -3, -2, 83, 27, 43, -1, 45, -3, -7, 39, -3, -2, -1, -5, 27, -3, 71, 39, -3, -2, 83, -3, 197, -13, 281, -2, -11, 83
Offset: 1
A091249 A014580-indices of primitive irreducible GF(2)[X]-polynomials.
2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 45, 48, 49, 50, 51, 52, 53, 56, 58, 61, 64, 65, 68, 70, 73, 75, 76, 77, 78, 80, 81, 83, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95
Offset: 1
Keywords
Links
A337442 Number of output sequences from the linear feedback shift register whose feedback polynomial coefficients (excluding the constant term) correspond to the binary representation of n.
1, 2, 3, 2, 4, 2, 2, 4, 6, 2, 4, 4, 2, 6, 4, 4, 8, 4, 2, 6, 2, 4, 8, 2, 4, 8, 4, 2, 6, 2, 2, 8, 14, 2, 6, 4, 8, 8, 4, 6, 6, 8, 12, 4, 4, 2, 8, 6, 2, 12, 8, 2, 8, 8, 2, 4, 4, 2, 4, 12, 6, 4, 6, 10, 20, 2, 4, 8, 2, 12, 6, 2, 2, 6, 4, 8, 16, 8, 2, 8, 4, 4, 16, 2
Offset: 0
Comments
a(n) > 1 for n > 0.
It appears that every term after a(2) is even.
It appears that a(2^n) is greater than each preceding term and is greater than or equal to each term up to a(2^(n+1)).
If a(n) = 2, then the nonzero shift register sequence is an m-sequence.
Examples
For n = 3 = 11 in binary, the polynomial is 1+x+x^2 and the 2 shift register sequences are {00..., 01101...}.
For n = 4 = 100 in binary, the polynomial is 1+x^3 and the 4 shift register sequences are {000..., 001001..., 011011..., 111...}.
For n = 6 = 110 in binary, the polynomial is 1+x^2+x^3 and the 2 shift register sequences are {000..., 0010111001...}.
For n = 10 = 1010 in binary, the polynomial is 1+x^2+x^4 and the 4 shift register sequences are {0000..., 0001010001..., 0011110011..., 0110110...}.
For n = 11 = 1011 in binary, the polynomial in 1+x+x^2+x^4. Using a Fibonacci LSFR, if the current state of the register is 0001, the next input bit is 0+0+1=1, and the next state is 0011. If the current state is 0100, the next input bit is 0+0+0=0, and the next state is 1000. The 4 shift register sequences are {0000..., 00011010001..., 00101110010..., 1111...}.
Comments
Examples
Links
Crossrefs