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A099895
XOR BINOMIAL transform of A000069 (Odious numbers).
1
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.
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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 29 2004
STATUS
approved