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
Examples
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.
Crossrefs
Programs
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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}
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
Comments