A099895 XOR BINOMIAL transform of A000069 (Odious numbers).

1, 3, 5, 0, 9, 0, 0, 0, 17, 0, 0, 0, 0, 0, 0, 0, 33, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 65, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 129

Offset: 0

Author

Paul D. Hanna, Oct 29 2004

Keywords

• nonn

Comments

See A099884 for the definitions of the XOR BINOMIAL transform and the XOR difference triangle.

Examples

			XOR difference triangle of A000069 begins:
[1],
[2,3],
[4,6,5],
[7,3,5,0],
[8,15,12,9,9],
[11,3,12,0,9,0],
[13,6,5,9,9,0,0],
[14,3,5,0,9,0,0,0],
[16,30,29,24,24,17,17,17,17],...
where A000069 is in the leftmost column,
and this sequence forms the main diagonal.
		

Crossrefs

Cf. A099884, A000069, A099896.

Programs

PARI

{a(n)=local(B);B=0;for(i=0,n,B=bitxor(B,binomial(n,i)%2*A000069(n-i) ));B}

Formula

a(2^n) = 2^(n+1)+1 for n>0, with a(0)=1 and a(k)=0 otherwise. a(n) = SumXOR_{i=0..n} (C(n, i)mod 2)*A000069(n-i) and SumXOR is summation under XOR.