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%I #36 Mar 10 2019 23:32:23
%S 0,1,1,0,1,2,-1,0,1,0,4,3,-2,-3,1,0,1,2,-3,-2,7,8,3,4,-3,-2,-7,-6,3,4,
%T -1,0,1,0,6,5,-9,-10,-4,-5,11,10,16,15,1,0,6,5,-4,-5,1,0,-14,-15,-9,
%U -10,6,5,11,10,-4,-5,1,0,1,2,-5,-4,16,17,10,11,-19
%N The bottom entry in the difference table of the binary digits of n.
%C By convention, a(0) = 0.
%C For any n > 0: let (b_0, ..., b_w) be the binary representation of n:
%C - b_w = 1, and for any k = 0..w, 0 <= b_k <= 1,
%C - n = Sum_{k = 0..w} b_k * 2^k,
%C - a(n) is the unique value remaining after taking successively the first differences of (b_0, ..., b_w) w times.
%C From _Robert Israel_, Mar 07 2019: (Start)
%C If n is odd then f(A030101(n)) = (-1)^A000523(n)*f(n).
%C In particular, if n is in A048701 then a(n)=0.
%C a(n) == 1 (mod A014963(A000523(n))) if n is even,
%C a(n) == 0 (mod A014963(A000523(n))) if n is odd. (End)
%H Robert Israel, <a href="/A306607/b306607.txt">Table of n, a(n) for n = 0..10000</a>
%F a(2^k) = 1 for any k >= 0.
%F a(2^k-1) = 0 for any k > 1.
%F a(3*2^k) = -k for any k >= 0.
%F a(n) = Sum_{k=0..A000523(n)} binomial(A000523(n), k)*(-1)^k*A030302(n,k). - _David A. Corneth_, Mar 07 2019
%F G.f.: 1/(x-1)*Sum_{k>=0}(x^(2^(k+1))-x^(2^k) + x^(2^k)/(x^(2^k)+1)*Sum_{m>=k+1}(binomial(m,k)*(-1)^(m-k)*(x^(2^(m+1))-x^(2^m)))). - _Robert Israel_, Mar 07 2019
%e For n = 42:
%e - the binary representation of 42 is "101010",
%e - the corresponding difference table is:
%e 0 1 0 1 0 1
%e 1 -1 1 -1 1
%e -2 2 -2 2
%e 4 -4 4
%e -8 8
%e 16
%e - hence a(42) = 16.
%p f:= proc(n) local L;
%p L:= convert(n,base,2);
%p while nops(L) > 1 do
%p L:= L[2..-1]-L[1..-2]
%p od;
%p op(L)
%p end proc:
%p map(f, [$0..100]); # _Robert Israel_, Mar 07 2019
%t a[n_] := NestWhile[Differences, Reverse[IntegerDigits[n, 2]], Length[#] > 1 &][[1]]; Array[a, 100, 0] (* _Amiram Eldar_, Mar 08 2019 *)
%o (PARI) a(n) = if (n, my (v=Vecrev(binary(n))); while (#v>1, v=vector(#v-1, k, (v[k+1]-v[k]))); v[1], 0)
%o (PARI) a(n) = my(b = binary(n), s = -1); sum(i = 1, #b, s=-s; binomial(#b-1, i-1) * b[i] * s) \\ _David A. Corneth_, Mar 07 2019
%Y Cf. A000523, A007088, A030101, A030190, A030302, A048701, A014963, A187202, A241494.
%K sign,base,look
%O 0,6
%A _Rémy Sigrist_, Feb 28 2019
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