A330602 a(n) = a(n-1) XOR (n+1), with a(0) = 0.

0, 2, 1, 5, 0, 6, 1, 9, 0, 10, 1, 13, 0, 14, 1, 17, 0, 18, 1, 21, 0, 22, 1, 25, 0, 26, 1, 29, 0, 30, 1, 33, 0, 34, 1, 37, 0, 38, 1, 41, 0, 42, 1, 45, 0, 46, 1, 49, 0, 50, 1, 53, 0, 54, 1, 57, 0, 58, 1, 61, 0, 62, 1, 65, 0, 66, 1, 69, 0, 70, 1, 73, 0, 74, 1, 77, 0, 78

Offset

0,2

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (0,1,0,1,0,-1).

Formula

a(n) = a(n-1) XOR (n+1), with a(0) = 0.

From Colin Barker, Dec 19 2019: (Start)

G.f.: x*(2 + x + 3*x^2 - x^3 - x^4) / ((1 - x)^2*(1 + x)^2*(1 + x^2)).

a(n) = a(n-2) + a(n-4) - a(n-6) for n>5.

(End)

From Stefano Spezia, Jun 20 2021: (Start)

E.g.f.: ((1 + 2*x)*cosh(x) - cos(x) - sin(x) + 3*sinh(x))/2.

a(n) = (2 + n - (-1)^n*(1 + n) - A057077(n))/2. (End)

Mathematica

a[0] = 0; a[n_] := a[n] = BitXor[a[n-1], n+1]; Array[a, 100, 0] (* Amiram Eldar, Dec 19 2019 *)
{0, #, 1, #+1}[[Mod[#, 4, 1]]]&/@Range@100 (* Federico Provvedi, May 11 2021 *)
LinearRecurrence[{0, 1, 0, 1, 0, -1}, {0, 2, 1, 5, 0, 6}, 80] (* Harvey P. Dale, Aug 07 2022 *)

Prog

(JavaScript) function generate (n) {
  let seq = [];
  for (let i = 1; i < n; i++) { seq.push(i) };
  let last = 0;
  return [0, ...seq.map(i => last = last ^ (i + 1))];
}
(PARI) concat(0, Vec(x*(2 + x + 3*x^2 - x^3 - x^4) / ((1 - x)^2*(1 + x)^2*(1 + x^2)) + O(x^70))) \\ Colin Barker, Dec 19 2019

Crossrefs

Bisections are: A000035 (even part), A042963(n+2) (odd part).

Cf. A057077.

Sequence in context: A021469 A090985 A011131 * A058241 A021827 A338554

Adjacent sequences: A330599 A330600 A330601 * A330603 A330604 A330605

Keyword

base,nonn,easy

Author

Kyle West, Dec 19 2019

Status

approved