A099891 XOR difference triangle of A003188 (Gray code numbers), read by rows.

0, 1, 1, 3, 2, 3, 2, 1, 3, 0, 6, 4, 5, 6, 6, 7, 1, 5, 0, 6, 0, 5, 2, 3, 6, 6, 0, 0, 4, 1, 3, 0, 6, 0, 0, 0, 12, 8, 9, 10, 10, 12, 12, 12, 12, 13, 1, 9, 0, 10, 0, 12, 0, 12, 0, 15, 2, 3, 10, 10, 0, 0, 12, 12, 0, 0, 14, 1, 3, 0, 10, 0, 0, 0, 12, 0, 0, 0, 10, 4, 5, 6, 6, 12, 12, 12, 12, 0, 0, 0, 0, 11, 1

Offset

0,4

Comments

Main diagonal is A099892, the XOR BINOMIAL transform of A003188. See A099884 for the definitions of the XOR BINOMIAL transform and the XOR difference triangle.

Links

Table of n, a(n) for n=0..92.

Formula

T(n, k) = SumXOR_{i=0..k} (C(k, i)mod 2)*(A003188(n-i)), where SumXOR is the analog of summation under the binary XOR operation and C(k, i)mod 2 = A047999(k, i). T(2^n, 2^n) = 3*2^(n-1) for n>0, with T(1, 1)=1 and T(k, k)=0 elsewhere.

T(n,1) = A006519(n), the lowest 1-bit of n (see formula by Franklin T. Adams-Watters in A003188). - Kevin Ryde, Jul 02 2020

Example

Rows begin:
[0],
[1,1],
[3,2,3],
[2,1,3,0],
[6,4,5,6,6],
[7,1,5,0,6,0],
[5,2,3,6,6,0,0],
[4,1,3,0,6,0,0,0],
[12,8,9,10,10,12,12,12,12],
...
where A003188 fills the leftmost column.

Prog

(PARI) {T(n, k)=local(B); B=0; for(i=0, k, B=bitxor(B, binomial(k, i)%2*(bitxor((n-i), (n-i)\2)))); B}

Crossrefs

Cf. A047999, A003188 (column k=0), A006519 (column k=1), A099892 (diagonal n=k).

Other triangles: A099884, A099889, A099893.

Sequence in context: A344220 A318056 A096839 * A241173 A096835 A237838

Adjacent sequences: A099888 A099889 A099890 * A099892 A099893 A099894

Keyword

nonn,tabl

Author

Paul D. Hanna, Oct 29 2004

Status

approved