cp's OEIS Frontend

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

%I A099902 #25 Jul 31 2025 02:16:35
%S A099902 1,3,7,11,23,59,103,139,279,827,1895,2955,5655,14395,24679,32907,
%T A099902 65815,197435,460647,723851,1512983,3881019,6774887,9142411,18219287,
%U A099902 54002491,123733863,192940939,369104407,939538491,1610637415,2147516555
%N A099902 Multiplies by 2 and shifts right under the XOR BINOMIAL transform (A099901).
%C A099902 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.
%C A099902 Bisection of A101624. - _Paul Barry_, May 10 2005
%H A099902 Robert Israel, <a href="/A099902/b099902.txt">Table of n, a(n) for n = 0..3290</a>
%F A099902 a(n) = SumXOR_{k=0..n} (binomial(n-k+floor(k/2), floor(k/2)) mod 2)*2^k for n >= 0.
%F A099902 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).
%F A099902 a(n) = Sum_{k=0..n} A047999(n-k+floor(k/2), floor(k/2)) * 2^k.
%F A099902 From _Paul Barry_, May 10 2005: (Start)
%F A099902 a(n) = Sum_{k=0..2n} (binomial(k, 2n-k) mod 2)*2^(2n-k);
%F A099902 a(n) = Sum_{k=0..n} (binomial(2n-k, k) mod 2)*2^k. (End)
%F A099902 a(n) = Sum_{k=0..2n} A106344(2n,k)*2^(2n-k). - _Philippe Deléham_, Dec 18 2008
%p A099902 a:= n -> add((binomial(n-k+floor(k/2),floor(k/2)) mod 2)*2^k, k=0..n):
%p A099902 map(a, [$0..100]); # _Robert Israel_, Jan 24 2016
%o A099902 (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}
%o A099902 (PARI) a(n)=sum(k=0,n,binomial(n-k+k\2,k\2)%2*2^k)
%o A099902 (Python)
%o A099902 def A099902(n): return sum(int(not ~((n<<1)-k)&k)<<k for k in range(n+1)) # _Chai Wah Wu_, Jul 30 2025
%Y A099902 Cf. A099884, A099900, A099901.
%K A099902 eigen,nonn
%O A099902 0,2
%A A099902 _Paul D. Hanna_, Oct 30 2004