A260022 -id:A260022 - cp's OEIS frontend

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

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, 5653987329, 9160491010

Offset: 0

Max Alekseyev, Nov 18 2009

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
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

G. C. Greubel, Table of n, a(n) for n = 0..1000
Karl-Heinz Hofmann and Frederik P.J. Vandecasteele, Terms in binary, including a visualization.

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. A000120, A000129, A206296, A248663, A002487.

a:= proc(n) option remember;
      `if`(n<2, n, Bits[Xor](2*a(n-1), a(n-2)))
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Jun 16 2025

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 *)

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

{ a168081(n) = subst(lift(polchebyshev(n-1,2,x/2)*Mod(1,2)),x,2); } \\ Max Alekseyev, Jun 20 2025

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

For n>1, a(n) = (2*a(n-1)) XOR a(n-2).
a(n) = A248663(A206296(n)). - Antti Karttunen, Dec 11 2015
A000120(a(n)) = A002487(n). - Karl-Heinz Hofmann, Jun 16 2025
a(n) = Sum_{k=0..n} (A049310(n,k) mod 2) * 2^k. - Max Alekseyev, Jun 20 2025

A006921 Diagonals of Pascal's triangle mod 2 interpreted as binary numbers.

Original entry on oeis.org

1, 1, 3, 2, 7, 5, 13, 8, 29, 21, 55, 34, 115, 81, 209, 128, 465, 337, 883, 546, 1847, 1301, 3357, 2056, 7437, 5381, 14087, 8706, 29443, 20737, 53505, 32768, 119041, 86273, 226051, 139778, 472839, 333061, 859405, 526344, 1903901, 1377557, 3606327

Offset: 0

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 0..1000
B. R. Hodgson, Letter to N. J. A. Sloane, Oct. 1991
B. R. Hodgson, On some number sequences related to the parity of binomial coefficients, Fib. Quart., 30 (1992), 35-47.

Cf. A011973, A000079, A047999 (Sierpiński), A007318, A101624.
Cf. A168081, A260022.
Cf. A257971 (first differences).

a006921 = sum . zipWith (*)
                a000079_list . map (flip mod 2) . reverse . a011973_row
-- Reinhard Zumkeller, Jul 14 2015

b2:=(n,k)->binomial(n,k) mod 2;
H:=n->add(b2(n-r,r)*2^( floor(n/2)-r ), r=0..floor(n/2));
[seq(H(n),n=0..30)]; # N. J. A. Sloane, Jul 14 2015

def A006921(n): return sum(int(not r & ~(n-r))*2**(n//2-r) for r in range(n//2+1)) # Chai Wah Wu, Jun 20 2022

a(2*n) = A260022(n); a(2*n+1) = A168081(n+1). - Reinhard Zumkeller, Jul 14 2015

a(n) = Sum_{r=0..n/2} binomial(n-r,r){mod 2} * 2^(floor(n/2)-r). - _N. J. A. Sloane, Jul 14 2015

cp's OEIS Frontend

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

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