OFFSET
1,3
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).
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)
EXAMPLE
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.
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul D. Hanna, Nov 05 2011
STATUS
approved