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XOR BINOMIAL transform of A000069 (Odious numbers).
1

%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