cp's OEIS Frontend

A168081 Lucas sequence U_n(x,1) over the field GF(2)[x].

%I A168081 #41 Jun 20 2025 10:44:54
%S A168081 0,1,2,5,8,21,34,81,128,337,546,1301,2056,5381,8706,20737,32768,86273,
%T A168081 139778,333061,526344,1377557,2228770,5308753,8388736,22085713,
%U A168081 35782690,85262357,134742024,352649221,570556418,1359020033,2147483648,5653987329,9160491010
%N A168081 Lucas sequence U_n(x,1) over the field GF(2)[x].
%C A168081 The Lucas sequence U_n(x,1) over the field GF(2)={0,1} is: 0, 1, x, x^2+1, x^3, x^4+x^2+1, x^5+x, ... Numerical values are obtained evaluating these 01-polynomials at x=2 over the integers.
%C A168081 The counterpart sequence is V_n(x,1) = x*U_n(x,1) that implies identities like U_{2n}(x,1) = x*U_n(x,1)^2. - _Max Alekseyev_, Nov 19 2009
%C A168081 Also, Chebyshev polynomials of the second kind evaluated at x=1/2 (A049310) with the resulting coefficients taken modulo 2, and then evaluated at x=2. - _Max Alekseyev_, Jun 20 2025
%H A168081 G. C. Greubel, <a href="/A168081/b168081.txt">Table of n, a(n) for n = 0..1000</a>
%H A168081 Karl-Heinz Hofmann and Frederik P.J. Vandecasteele, <a href="/A168081/a168081_1.txt">Terms in binary, including a visualization</a>.
%F A168081 For n>1, a(n) = (2*a(n-1)) XOR a(n-2).
%F A168081 a(n) = A248663(A206296(n)). - _Antti Karttunen_, Dec 11 2015
%F A168081 A000120(a(n)) = A002487(n). - _Karl-Heinz Hofmann_, Jun 16 2025
%F A168081 a(n) = Sum_{k=0..n} (A049310(n,k) mod 2) * 2^k. - _Max Alekseyev_, Jun 20 2025
%p A168081 a:= proc(n) option remember;
%p A168081       `if`(n<2, n, Bits[Xor](2*a(n-1), a(n-2)))
%p A168081     end:
%p A168081 seq(a(n), n=0..35);  # _Alois P. Heinz_, Jun 16 2025
%t A168081 a[0] = 0; a[1] = 1; a[n_] := a[n] = BitXor[2 a[n - 1], a[n - 2]]; Table[a@ n, {n, 0, 32}] (* _Michael De Vlieger_, Dec 11 2015 *)
%o A168081 (PARI) { a=0; b=1; for(n=1,50, c=bitxor(2*b,a); a=b; b=c; print1(c,", "); ); }
%o A168081 (Python)
%o A168081 def A168081(n): return sum(int(not r & ~(2*n-1-r))*2**(n-1-r) for r in range(n)) # _Chai Wah Wu_, Jun 20 2022
%o A168081 (PARI) { a168081(n) = subst(lift(polchebyshev(n-1,2,x/2)*Mod(1,2)),x,2); } \\ _Max Alekseyev_, Jun 20 2025
%Y A168081 A bisection of A006921. Cf. A260022. - _N. J. A. Sloane_, Jul 14 2015
%Y A168081 See also A257971, first differences of A006921. - _Reinhard Zumkeller_, Jul 14 2015
%Y A168081 Cf. A000120, A000129, A206296, A248663, A002487.
%K A168081 nonn,base
%O A168081 0,3
%A A168081 _Max Alekseyev_, Nov 18 2009