A099894 XOR BINOMIAL transform of A038712.

1, 2, 0, 4, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 32, 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, 64, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Author

Paul D. Hanna, Oct 29 2004

Keywords

• nonn

Comments

See A099884 for the definitions of the XOR BINOMIAL transform and the XOR difference triangle.

a(n) = A062383(n+1) - A062383(n). - Reinhard Zumkeller, Aug 06 2009

A038712 has offset 1, but we need to use offset 0 for the XOR BINOMIAL. - Michael Somos, Dec 30 2016

Examples

			G.f. = 1 + 2*x + 4*x^3 + 8*x^7 + 16*x^15 + 32*x^31 + 64*x^63 + 128*x^127 + ...
XOR difference triangle of A038712 begins:
[1],
[3,2],
[1,2,0],
[7,6,4,4],
[1,6,0,4,0],
[3,2,4,4,0,0],
[1,2,0,4,0,0,0],
[15,14,12,12,8,8,8,8],...
where A038712 is in the leftmost column and A099894 (this sequence) forms the main diagonal.
a(1) = 1*1 XOR 0*1 = 1, a(2) = 1*1 XOR 0*3 XOR 1*1 = 0, a(3) = 1*1 XOR 1*3 XOR 1*1 XOR 1*7 = 4 where (1, 3, 1, 7) are the first four terms of A038712. - _Michael Somos_, Dec 30 2016
		

Crossrefs

Cf. A099884, A038712, A099895, A062383, A048298.

Programs

Mathematica

a[ n_] := With[ {m = n+1}, If[ m >=0 && Total[ IntegerDigits[ m, 2]] == 1, m, 0]]; (* Michael Somos, Dec 30 2016 *)

PARI

{a(n)=local(B);B=0;for(i=0,n,B=bitxor(B,binomial(n,i)%2*A038712(n-i) ));B}

PARI

{a(n) = my(m = n+1); m * ( m>=0 && hammingweight(m) == 1)}; /* Michael Somos, Dec 30 2016 */

Formula

a(2^n-1) = 2^n for n>=0 and a(k)=0 otherwise. a(n) = SumXOR_{i=0..n} (C(n, i)mod 2)*A038712(n-i) and SumXOR is summation under XOR.

a(n) = A048298(n+1). - Michael Somos, Dec 30 2016