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%I #9 May 11 2014 22:53:30
%S 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,
%T 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
%N XOR BINOMIAL transform of A000069 (Odious numbers).
%C See A099884 for the definitions of the XOR BINOMIAL transform and the XOR difference triangle.
%F 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.
%e XOR difference triangle of A000069 begins:
%e [1],
%e [2,3],
%e [4,6,5],
%e [7,3,5,0],
%e [8,15,12,9,9],
%e [11,3,12,0,9,0],
%e [13,6,5,9,9,0,0],
%e [14,3,5,0,9,0,0,0],
%e [16,30,29,24,24,17,17,17,17],...
%e where A000069 is in the leftmost column,
%e and this sequence forms the main diagonal.
%o (PARI) {a(n)=local(B);B=0;for(i=0,n,B=bitxor(B,binomial(n,i)%2*A000069(n-i) ));B}
%Y Cf. A099884, A000069, A099896.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Oct 29 2004