A003815 a(0) = 0, a(n) = a(n-1) XOR n.

0, 1, 3, 0, 4, 1, 7, 0, 8, 1, 11, 0, 12, 1, 15, 0, 16, 1, 19, 0, 20, 1, 23, 0, 24, 1, 27, 0, 28, 1, 31, 0, 32, 1, 35, 0, 36, 1, 39, 0, 40, 1, 43, 0, 44, 1, 47, 0, 48, 1, 51, 0, 52, 1, 55, 0, 56, 1, 59, 0, 60, 1, 63, 0, 64, 1, 67, 0

Offset: 0

Author

Marc LeBrun

Keywords

• nonn
• base

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0, 1, 0, 1, 0, -1).

Crossrefs

Cf. A003816.

Cf. A077140, A145768. - M. F. Hasler, Oct 20 2008

Programs

Mathematica

an = 0; Reap[ For[i = 0, i <= 100, i++, an = BitXor[an, i]; Sow[an]]][[2, 1]] (* Jean-François Alcover, Oct 11 2013, translated from PARI *)
CoefficientList[Series[x (1 + 3 x - x^2 + x^3)/((1 - x^4) (1 - x^2)), {x, 0, 100}], x] (* Vincenzo Librandi, Oct 12 2013 *)
nxt[{n_,a_}]:={n+1,BitXor[n+1,a]}; NestList[nxt,{0,0},70][[All,2]] (* Harvey P. Dale, Mar 10 2019 *)
{#,1,#+1,0}[[1+Mod[#,4]]]&/@Range[0,100] (* Federico Provvedi, May 10 2021 *)

PARI

print1(an=0); for( i=1,100, print1(",",an=bitxor(an,i))) \\ M. F. Hasler, Oct 20 2008

Formula

a(n) = n + (-1)^n*a(n-1). - Vladeta Jovovic, Mar 13 2003

a(0)=0, a(4n+1)=1, a(4n+2)=4n+3, a(4n+3)=0, a(4n+4)=4n+4, n >= 0.

a(n) = f(n,0) with f(n,x) = x if n=0, otherwise f(n-1,x+n) if x is even, otherwise f(n-1,x-n). - Reinhard Zumkeller, Oct 09 2007

a(n) = abs(A077140(n)) for n > 0. - Reinhard Zumkeller, Oct 09 2007

G.f.: x*(1+3*x-x^2+x^3)/((1-x^4)*(1-x^2)). - Vincenzo Librandi, Oct 12 2013

a(n) = (1 + n + n*(-1)^n + (-1)^floor((n-1)/2))/2. - Wesley Ivan Hurt, May 08 2021