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%I #5 Mar 03 2024 17:49:04
%S 1,2,3,2,4,4,6,4,6,8,6,4,8,8,12,8,10,16,6,4,8,8,12,8,12,16,12,8,16,16,
%T 24,16,18,32,6,4,8,8,12,8,12,16,12,8,16,16,24,16,20,32,12,8,16,16,24,
%U 16,24,32,24,16,32,32,48,32,34,64,6,4,8,8,12,8,12,16,12,8,16,16,24,16
%N Number of values of k for which C(n,k)-C(n-2,k-1) is odd.
%F If 2*2^m+1<n<=3*2^m+1 then a(n)=a(n-2^m) except that a(3*2^m)=a(2^m)+2; if 3*2^m+1<n<=4*2^m+1 then a(n)=2*a(n-2^m) except that a(4*2^m)=2*a(3*2^m)-6. So a(2^m)=2^(m-1)+2, a(2^m+1)=2^m, a(2^m+2)=6, a(2^m+3)=4, a(3*2^m)=2^m+4, a(3*2^m+1)=2^(m+1), a(3*2^m+2)=12, a(3*2^m+3)=8, etc. for m large enough to avoid confusion.
%e a(5)=4 since the values of C(5,k)-C(3,k-1) are 1-0, 5-1, 10-3, 10-3, 5-1, 1-0, i.e. 1,4,7,7,4,1 of which 4 are odd.
%t Join[{1,2},Table[Count[Table[Binomial[n,k]-Binomial[n-2,k-1],{k,0,n}],_?OddQ],{n,2,80}]] (* _Harvey P. Dale_, Mar 03 2024 *)
%Y Cf. A001316, A072405.
%Y Cf. A094959.
%K nonn
%O 0,2
%A _Henry Bottomley_, Jun 16 2002
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