A003816 a(0) = 0, a(n) = a(n-1) XOR -n.

0, 1, -1, 4, 0, 5, -1, 8, 0, 9, -1, 12, 0, 13, -1, 16, 0, 17, -1, 20, 0, 21, -1, 24, 0, 25, -1, 28, 0, 29, -1, 32, 0, 33, -1, 36, 0, 37, -1, 40, 0, 41, -1, 44, 0, 45, -1, 48, 0, 49, -1, 52, 0, 53, -1, 56, 0, 57, -1, 60, 0, 61, -1, 64, 0, 65, -1, 68

Offset: 0

Author

Marc LeBrun

Keywords

• sign
• easy

Links

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

Crossrefs

Cf. A003815.

Programs

Magma

[(n-(n+1)*(-1)^n+(-1)^((2*n+1-(-1)^n) div 4))/2 : n in [0..100]]; // Wesley Ivan Hurt, May 03 2016

Maple

A003816:=n->(n-(n+1)*(-1)^n+(-1)^((2*n+1-(-1)^n)/4))/2: seq(A003816(n), n=0..150); # Wesley Ivan Hurt, May 03 2016

Mathematica

CoefficientList[Series[x*(1 - x + 3 x^2 + x^3)/((x^2 - 1)^2*(x^2 + 1)), {x, 0, 100}], x] (* Wesley Ivan Hurt, May 01 2016 *)
nxt[{n_,a_}]:={n+1,BitXor[a,-n-1]}; -#&/@(NestList[nxt,{0,0},70][[All,2]]) (* or *) LinearRecurrence[{0,1,0,1,0,-1},{0,1,-1,4,0,5},70] (* Harvey P. Dale, Oct 16 2019 *)

PARI

a(n)=([0,1,0,0,0,0; 0,0,1,0,0,0; 0,0,0,1,0,0; 0,0,0,0,1,0; 0,0,0,0,0,1; -1,0,1,0,1,0]^n*[0;1;-1;4;0;5])[1,1] \\ Charles R Greathouse IV, May 02 2016

Formula

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

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

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

From Wesley Ivan Hurt, May 01 2016, May 03 2016: (Start)

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

a(n) = (-1)^floor((n-1)/2) * Sum_{i=0..n} i*(-1)^floor(i/2).

a(n) = (n-(n+1)*(-1)^n+(-1)^((2*n+1-(-1)^n)/4))/2. (End)

E.g.f.: (cos(x) + (-1 + 2*x)*cosh(x) - sin(x) + sinh(x))/2. - Ilya Gutkovskiy, May 03 2016