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

0,3

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..85.

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

Mathematica

a[n_] := Module[{e = IntegerExponent[n, 2]}, If[n == 2^e, 3*2^(e-1), 0]]; Array[a, 100, 0] (* Amiram Eldar, Aug 31 2023 *)

Prog

(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}

Crossrefs

Cf. A099884, A003188, A099891.

Sequence in context: A009780 A175677 A129502 * A060147 A268142 A321254

Adjacent sequences: A099889 A099890 A099891 * A099893 A099894 A099895

Keyword

nonn,easy,mult

Author

Paul D. Hanna, Oct 29 2004

Status

approved