A099889 -id:A099889 - cp's OEIS frontend

A099890 XOR BINOMIAL transform of the odd numbers; also the main diagonal of the XOR difference triangle A099889.

1, 2, 4, 0, 8, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 32, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 128

Offset: 0

Paul D. Hanna, Oct 29 2004

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..1024

Cf. A099884, A099889.

Join[{1},Table[If[IntegerQ[Log2[n]],2n,0],{n,512}]] (* or *) Join[{1},With[{kk=10},Flatten[ Riffle[ 2^Range[kk],Table[PadRight[{},2^k-1,0],{k,0,kk}]]]]] (* Harvey P. Dale, Aug 11 2024 *)

a(n)=local(B);B=0;for(i=0,n,B=bitxor(B,binomial(n,i)%2*(2*(n-i)+1)));B

a(0) = 1; a(2^n) = 2^(n+1); a(n) = 0 otherwise.

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

Original entry on oeis.org

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

Paul D. Hanna, Oct 29 2004

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.

			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.

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

Other triangles: A099884, A099889, A099893.

{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}

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

cp's OEIS Frontend

A099890 XOR BINOMIAL transform of the odd numbers; also the main diagonal of the XOR difference triangle A099889.

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