A099901 Shifts left and divides by 2 under the XOR BINOMIAL transform (A099902).

1, 2, 6, 14, 22, 46, 118, 206, 278, 558, 1654, 3790, 5910, 11310, 28790, 49358, 65814, 131630, 394870, 921294, 1447702, 3025966, 7762038, 13549774, 18284822, 36438574, 108004982, 247467726, 385881878, 738208814, 1879076982, 3221274830

Offset: 0

Author

Paul D. Hanna, Oct 30 2004

Keywords

• eigen
• nonn

Comments

Equals the XOR BINOMIAL transform of A099902. Also, equals the leftmost column of the XOR difference triangle A099900, in which the central terms of the rows forms the powers of 2.

Crossrefs

Cf. A099884, A099900, A099902.

Programs

PARI

{a(n)=local(B);B=0;if(n==0,B=1, for(k=0,n-1, B=bitxor(B,binomial(n-1-k+k\2,k\2)%2*2^(k+1))));B}

Formula

a(0)=1; a(n) = SumXOR_{k=0..n-1} (C(n-1-k+[k/2], [k/2])mod 2)*2^(k+1) for n>0. a(n) = SumXOR_{i=0..n} (C(n, i)mod 2)*A099902(n-i), where SumXOR is the analog of summation under the binary XOR operation and C(k, i)mod 2 = A047999(k, i).