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A099891
XOR difference triangle of A003188 (Gray code numbers), read by rows.
1
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.
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
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Oct 29 2004
STATUS
approved