A099892 XOR BINOMIAL transform of A003188 (Gray code numbers); also the main diagonal of the XOR difference triangle A099891.
0, 1, 3, 0, 6, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 48, 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, 96, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 0
Programs
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Mathematica
a[n_] := Module[{e = IntegerExponent[n, 2]}, Switch[n, 0, 0, 1, 1, 2^e, 3*2^(e - 1), , 0]]; Array[a, 100, 0] (* _Amiram Eldar, Aug 31 2023, corrected by Michael Shamos, May 22 2025 *) -
PARI
{a(n)=local(B);B=0;for(i=0,n,B=bitxor(B,binomial(n,i)%2*(bitxor((n-i),(n-i)\2))));B}
Formula
a(2^n) = 3*2^(n-1) for n>0, with a(0)=0, a(1) = 1 and a(k)=0 otherwise. a(n) = SumXOR_{i=0..n} (C(n, i)mod 2)*A003188(n-i), where A003188(k)=bitxor(k, [k/2]) and SumXOR is summation under XOR.
Multiplicative with a(2^e) = 3*2^(e-1), a(p^e) = 0 otherwise. - David W. Wilson, Jun 12 2005
Dirichlet g.f.: (2^s+1)/(2^s-2). - R. J. Mathar, Apr 14 2011
Comments