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a(n) = s(n)*r - s(2^r + n), where s(n) = A002487(n) starting at n = 0 and r = 1 + floor(log_2(n)).
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%I #20 Dec 19 2018 18:45:33

%S -1,-1,0,1,1,4,3,5,2,9,7,12,5,13,8,11,3,16,13,23,10,27,17,24,7,25,18,

%T 29,11,26,15,19,4,25,21,38,17,47,30,43,13,48,35,57,22,53,31,40,9,41,

%U 32,55,23,60,37,51,14,47,33,52,19,43,24,29,5,36,31,57,26,73,47

%N a(n) = s(n)*r - s(2^r + n), where s(n) = A002487(n) starting at n = 0 and r = 1 + floor(log_2(n)).

%C The same sequence arises for any integer value r > 1 + floor(log_2(n)).

%C The maximum value of a(n) between n=2^k and n=2^(k+1) is the k-th term of A023619.

%F a(0)=-1; a(1)=-1;

%F a(2n) = a(n) + s(n);

%F a(2n+1) = a(n) + a(n+1) + s(2n+1).

%F a(n) = A156140(n) - A002487(n).

%t s[0] = 0; s[1] = 1; s[n_?EvenQ] := s[n] = s[n/2]; s[n_?OddQ] := s[n] = s[(n + 1)/2] + s[(n - 1)/2];

%t a[0]=-1; a[1]=-1; a[n_?EvenQ] := a[n] = a[n/2]+s[n/2]; a[n_?OddQ] := a[n] = s[(n + 1)/2] + s[(n - 1)/2]+a[(n + 1)/2] + a[(n - 1)/2]

%K sign

%O 0,6

%A _Aubrey Laskowski_, Oct 28 2018