A099899 Multiplies by 4 and shifts right under the XOR BINOMIAL transform (A099898).

1, 5, 21, 69, 277, 1349, 5141, 16453, 65813, 329029, 1381397, 4538437, 18088213, 88081733, 335549461, 1073758277, 4295033109, 21475165509, 90195694613, 296357281861, 1189724029205, 5793998964037, 22080762418197, 70666170679365

Offset: 0

Author

Paul D. Hanna, Oct 30 2004

Keywords

• nonn

Comments

Equals the XOR BINOMIAL transform of A099898. Also, equals the main diagonal of the XOR difference triangle A099897, in which the central terms of the rows form the powers of 4. See A099884 for the definitions of XOR difference triangle and the XOR BINOMIAL transform.

Crossrefs

Cf. A099884, A099897, A099898.

Programs

PARI

{a(n)=local(B);B=0;for(k=0,n,B=bitxor(B,binomial(n-k+k\2,k\2)%2*4^k));B}

Formula

a(n) = SumXOR_{k=0..n} (C(n-k+[k/2], [k/2])mod 2)*2^k for n>=0. a(n) = SumXOR_{i=0..n} (C(n, i)mod 2)*A099898(n-i), where SumXOR is the analog of summation under the binary XOR operation and C(i, j)mod 2 = A047999(i, j).