cp's OEIS Frontend

A306754 The bottom entry in the difference table of the positions of the ones in the binary representation of n.

%I A306754 #18 Mar 09 2019 00:24:32
%S A306754 0,0,1,1,2,2,1,0,3,3,2,1,1,-1,0,0,4,4,3,2,2,0,1,1,1,-2,-1,-2,0,1,0,0,
%T A306754 5,5,4,3,3,1,2,2,2,-1,0,-1,1,2,1,1,1,-3,-2,-4,-1,-1,-2,-3,0,2,1,3,0,
%U A306754 -1,0,0,6,6,5,4,4,2,3,3,3,0,1,0,2,3,2,2,2,-2
%N A306754 The bottom entry in the difference table of the positions of the ones in the binary representation of n.
%C A306754 By convention, a(0) = 0.
%C A306754 For any n > 0:
%C A306754 - let (b_1, b_2, ..., b_h) be the positions of the ones in the binary representation of n,
%C A306754 - h = A000120(n) and 0 <= b_1 < b_2 < ... < b_h,
%C A306754 - n = Sum_{k = 1..h} 2^b_k,
%C A306754 - a(n) is the unique value remaining after taking successively the first differences of (b_1, ..., b_h) h-1 times.
%H A306754 Rémy Sigrist, <a href="/A306754/b306754.txt">Table of n, a(n) for n = 0..16384</a>
%F A306754 a(2^k) = k for any k >= 0.
%F A306754 a(2^k-1) = [k=2].
%F A306754 a(2*n) = a(n) + A209229(n).
%e A306754 For n = 59:
%e A306754 - the binary representation of 59 is "111011",
%e A306754 - so h = 5 and b_1 = 0, b_2 = 1, b_3 = 3, b_4 = 4, b_5 = 5,
%e A306754 - the corresponding difference table is:
%e A306754         0   1   3   4   5
%e A306754           1   2   1   1
%e A306754             1  -1   0
%e A306754              -2   1
%e A306754                 3
%e A306754 - hence a(59) = 3.
%o A306754 (PARI) a(n) = { my (h=hammingweight(n), o=0, v=0); forstep (k=h-1, 0, -1, my (w=valuation(n, 2)); o += w; v += (-1)^k * binomial(h-1, k) * o; o++; n\=2^(1+w)); v };
%Y A306754 See A306607 for a similar sequence.
%Y A306754 Cf. A000120, A133457, A209229.
%K A306754 sign,base
%O A306754 0,5
%A A306754 _Rémy Sigrist_, Mar 08 2019