cp's OEIS Frontend

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.

%I A337442 #18 Oct 14 2020 10:59:40
%S A337442 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,
%T A337442 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,
%U A337442 4,8,2,12,6,2,2,6,4,8,16,8,2,8,4,4,16,2
%N 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.
%C A337442 a(n) > 1 for n > 0.
%C A337442 It appears that every term after a(2) is even.
%C A337442 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)).
%C A337442 If a(n) = 2, then the nonzero shift register sequence is an m-sequence.
%e A337442 For n = 3 = 11 in binary, the polynomial is 1+x+x^2 and the 2 shift register sequences are {00..., 01101...}.
%e A337442 For n = 4 = 100 in binary, the polynomial is 1+x^3 and the 4 shift register sequences are {000..., 001001..., 011011..., 111...}.
%e A337442 For n = 6 = 110 in binary, the polynomial is 1+x^2+x^3 and the 2 shift register sequences are {000..., 0010111001...}.
%e A337442 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...}.
%e A337442 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...}.
%Y A337442 a(2^n) = A000031(n+1).
%Y A337442 A011260 counts how many 2's are in the interval [2^(n-1),(2^n)-1].
%Y A337442 a(n) = 2 if and only if 2n+1 is in A091250.
%Y A337442 Cf. A100447, A001037, A000016, A000013 (definition 2), A000020, A058947.
%Y A337442 Cf. A011655..A011751 for examples of binary m-sequences.
%K A337442 nonn,base
%O A337442 0,2
%A A337442 _Michael Schwartz_, Aug 27 2020