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

0, 1, 2, 5, 8, 21, 34, 81, 128, 337, 546, 1301, 2056, 5381, 8706, 20737, 32768, 86273, 139778, 333061, 526344, 1377557, 2228770, 5308753, 8388736, 22085713, 35782690, 85262357, 134742024, 352649221, 570556418, 1359020033, 2147483648

Offset

0,3

Comments

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.

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

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

For n>1, a(n) = (2*a(n-1)) XOR a(n-2).

a(n) = A248663(A206296(n)). - Antti Karttunen, Dec 11 2015

Mathematica

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 *)

Prog

(PARI) { a=0; b=1; for(n=1, 50, c=bitxor(2*b, a); a=b; b=c; print1(c, ", "); ) }
(Python)
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

Crossrefs

A bisection of A006921. Cf. A260022. - N. J. A. Sloane, Jul 14 2015

See also A257971, first differences of A006921. - Reinhard Zumkeller, Jul 14 2015

Cf. A000129, A206296, A248663.

Sequence in context: A092446 A087077 A200276 * A340399 A117647 A121568

Adjacent sequences: A168078 A168079 A168080 * A168082 A168083 A168084

Keyword

nonn

Author

Max Alekseyev, Nov 18 2009

Status

approved