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

0,2

Comments

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

Links

Table of n, a(n) for n=0..64.

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.

Example

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.

Prog

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

Crossrefs

Cf. A099884, A000069, A099896.

Sequence in context: A346254 A021289 A200480 * A248192 A323987 A124222

Adjacent sequences: A099892 A099893 A099894 * A099896 A099897 A099898

Keyword

nonn

Author

Paul D. Hanna, Oct 29 2004

Status

approved