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

0,2

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

Links

Table of n, a(n) for n=0..85.

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

Example

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

Mathematica

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

Prog

(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 */

Crossrefs

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

Sequence in context: A305212 A104774 A087263 * A048298 A123565 A258701

Adjacent sequences: A099891 A099892 A099893 * A099895 A099896 A099897

Keyword

nonn

Author

Paul D. Hanna, Oct 29 2004

Status

approved