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%I #20 Jun 20 2022 13:16:14
%S 0,1,2,5,8,21,34,81,128,337,546,1301,2056,5381,8706,20737,32768,86273,
%T 139778,333061,526344,1377557,2228770,5308753,8388736,22085713,
%U 35782690,85262357,134742024,352649221,570556418,1359020033,2147483648
%N Lucas sequence U_n(x,1) over the field GF(2).
%C 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 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
%H G. C. Greubel, <a href="/A168081/b168081.txt">Table of n, a(n) for n = 0..1000</a>
%F For n>1, a(n) = (2*a(n-1)) XOR a(n-2).
%F a(n) = A248663(A206296(n)). - _Antti Karttunen_, Dec 11 2015
%t 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 (PARI) { a=0; b=1; for(n=1,50, c=bitxor(2*b,a); a=b; b=c; print1(c,", "); ) }
%o (Python)
%o 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
%Y A bisection of A006921. Cf. A260022. - _N. J. A. Sloane_, Jul 14 2015
%Y See also A257971, first differences of A006921. - _Reinhard Zumkeller_, Jul 14 2015
%Y Cf. A000129, A206296, A248663.
%K nonn
%O 0,3
%A _Max Alekseyev_, Nov 18 2009
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