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A379719 a(0) = 0, and for any n > 0, a(n) is the least integer (in absolute value) not yet in the sequence such that the absolute difference of a(n-1) and a(n) is a power of 2; in case of a tie, preference is given to the positive value.

%I A379719 #16 Jan 18 2025 09:10:57
%S A379719 0,1,-1,-2,2,3,4,-4,-3,5,6,7,8,-8,-6,-5,-7,9,10,11,12,13,14,15,16,-16,
%T A379719 -12,-10,-9,-11,-13,-14,-15,17,18,19,20,21,22,23,24,25,26,27,28,29,30,
%U A379719 31,32,-32,-24,-20,-18,-17,-19,-21,-22,-23,-25,-26,-27,-28
%N A379719 a(0) = 0, and for any n > 0, a(n) is the least integer (in absolute value) not yet in the sequence such that the absolute difference of a(n-1) and a(n) is a power of 2; in case of a tie, preference is given to the positive value.
%C A379719 This sequence is a variant of A377091, based on powers of 2 instead of squares.
%C A379719 Every integer (positive or negative) appears in this sequence.
%C A379719 This sequence has indeed the following structure:
%C A379719 - a transient block T corresponding to the initial terms a(0) to a(8),
%C A379719 - then, for k = 2, 3, etc., blocks B(k) with the following features:
%C A379719 - the initial blocks T, B(2), ..., B(k-1) form a permutation of -2^k..2^k and end with the value -2^k + 1,
%C A379719 - the block B(k) starts with the positive values 2^k+1, 2^k+2, ..., 2^(k+1),
%C A379719 - then continues with the negative values -2^(k+1), -2^(k+1) + 2^(k-1), -2^(k+1) + 2^(k-1) + 2^(k-2), ..., -2^(k+1) + 2^(k-1) + 2^(k-2) + ... + 2^0,
%C A379719 - then continues with the missing negative values down to -2^(k+1) + 1 with steps of -1 or -2.
%C A379719 As a consequence, nonnegative values appear in natural order.
%H A379719 Rémy Sigrist, <a href="/A379719/b379719.txt">Table of n, a(n) for n = 0..10000</a>
%H A379719 Rémy Sigrist, <a href="/A379719/a379719.gp.txt">PARI program</a>
%e A379719 The first terms are:
%e A379719   n   a(n)  |a(n)-a(n-1)|
%e A379719   --  ----  -------------
%e A379719    0     0  N/A
%e A379719    1     1  2^0
%e A379719    2    -1  2^1
%e A379719    3    -2  2^0
%e A379719    4     2  2^2
%e A379719    5     3  2^0
%e A379719    6     4  2^0
%e A379719    7    -4  2^3
%e A379719    8    -3  2^0
%e A379719    9     5  2^3
%e A379719   10     6  2^0
%e A379719   11     7  2^0
%e A379719   12     8  2^0
%e A379719   13    -8  2^4
%e A379719   14    -6  2^1
%o A379719 (PARI) \\ See Links section.
%Y A379719 Cf. A377091, A377092.
%K A379719 sign
%O A379719 0,4
%A A379719 _Rémy Sigrist_, Dec 31 2024