A099902 Multiplies by 2 and shifts right under the XOR BINOMIAL transform (A099901).

1, 3, 7, 11, 23, 59, 103, 139, 279, 827, 1895, 2955, 5655, 14395, 24679, 32907, 65815, 197435, 460647, 723851, 1512983, 3881019, 6774887, 9142411, 18219287, 54002491, 123733863, 192940939, 369104407, 939538491, 1610637415, 2147516555

Offset

0,2

Comments

Equals the XOR BINOMIAL transform of A099901. Also, equals the main diagonal of the XOR difference triangle A099900, in which the central terms of the rows form the powers of 2.

Bisection of A101624. - Paul Barry, May 10 2005

Links

Robert Israel, Table of n, a(n) for n = 0..3290

Formula

a(n) = SumXOR_{k=0..n} (binomial(n-k+floor(k/2), floor(k/2)) mod 2)*2^k for n >= 0.

a(n) = SumXOR_{i=0..n} (C(n, i) mod 2)*A099901(n-i), where SumXOR is the analog of summation under the binary XOR operation and C(i, j) mod 2 = A047999(i, j).

a(n) = Sum_{k=0..n} A047999(n-k+floor(k/2), floor(k/2)) * 2^k.

From Paul Barry, May 10 2005: (Start)

a(n) = Sum_{k=0..2n} (binomial(k, 2n-k) mod 2)*2^(2n-k);

a(n) = Sum_{k=0..n} (binomial(2n-k, k) mod 2)*2^k. (End)

a(n) = Sum_{k=0..2n} A106344(2n,k)*2^(2n-k). - Philippe Deléham, Dec 18 2008

Maple

a:= n -> add((binomial(n-k+floor(k/2), floor(k/2)) mod 2)*2^k, k=0..n):
map(a, [$0..100]); # Robert Israel, Jan 24 2016

Prog

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

Crossrefs

Cf. A099884, A099900, A099901.

Sequence in context: A188132 A139814 A368943 * A336897 A316962 A360297

Adjacent sequences: A099899 A099900 A099901 * A099903 A099904 A099905

Keyword

eigen,nonn

Author

Paul D. Hanna, Oct 30 2004

Status

approved