A199403 Binary XOR of (2^k - (-1)^k)/3 as k varies from 1 to n.

1, 0, 3, 6, 13, 24, 51, 102, 205, 408, 819, 1638, 3277, 6552, 13107, 26214, 52429, 104856, 209715, 419430, 838861, 1677720, 3355443, 6710886, 13421773, 26843544, 53687091, 107374182, 214748365, 429496728, 858993459, 1717986918, 3435973837, 6871947672, 13743895347

Offset: 1

Author

Paul D. Hanna, Nov 05 2011

Keywords

• nonn
• easy

Examples

			a(2) = (2^1+1)/3 XOR (2^2-1)/3 = 1 XOR 1 = 0;
a(3) = (2^1+1)/3 XOR (2^2-1)/3 XOR (2^3+1)/3 = 1 XOR 1 XOR 3 = 3;
a(4) = (2^1+1)/3 XOR (2^2-1)/3 XOR (2^3+1)/3 XOR (2^4-1)/3 = 1 XOR 1 XOR 3 XOR 5 = 6.
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (2,0,0,1,-2).

Crossrefs

Cf. A199402, A199396.

Programs

Maple

a:= n-> (<<0|1|0|0|0>, <0|0|1|0|0>, <0|0|0|1|0>, <0|0|0|0|1>, <-2|1|0|0|2>>^n. <<0, 1, 0, 3, 6>>)[1, 1]: seq(a(n), n=1..60); # Alois P. Heinz, Nov 05 2011

Mathematica

FoldList[BitXor, Table[(2^n - (-1)^n)/3, {n, 1, 20}]] (* Vladimir Reshetnikov, Nov 02 2015 *)
Table[(6*Cos[Pi n/2] + 2*Sin[Pi n/2] + 4*2^n - 5*(-1)^n - 5)/10, {n, 1, 20}] (* Vladimir Reshetnikov, Nov 02 2015 *)

PARI

{a(n)=if(n<0,0,bitxor(a(n-1),((2^n-(-1)^n)/3)))}

PARI

Vec(x*(3*x^2-2*x+1)/((x-1)*(x+1)*(2*x-1)*(x^2+1)) + O(x^100)) \\ Colin Barker, Nov 02 2015

Formula

G.f.: (3*x^2-2*x+1)*x/(2*x^5-x^4-2*x+1). - Alois P. Heinz, Nov 05 2011

From Vladimir Reshetnikov, Nov 02 2015: (Start)

a(n) = (6*cos(Pi*n/2) + 2*sin(Pi*n/2) + 4*2^n - 5*(-1)^n - 5)/10.

Recurrence: a(1) = 1, a(2) = 0, a(3) = 3, a(4) = 6, a(5) = 13, a(n) = 2*a(n-1) + a(n-4) - 2*a(n-5).

E.g.f.: (2*cosh(2*x) - 5*cosh(x) + 2*sinh(2*x) + 3*cos(x) + sin(x))/5.

(End)