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.

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, -12, -10, -9, -11, -13, -14, -15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, -32, -24, -20, -18, -17, -19, -21, -22, -23, -25, -26, -27, -28

Offset: 0

Author

Rémy Sigrist, Dec 31 2024

Keywords

• sign

Comments

This sequence is a variant of A377091, based on powers of 2 instead of squares.

Every integer (positive or negative) appears in this sequence.

This sequence has indeed the following structure:

- a transient block T corresponding to the initial terms a(0) to a(8),

- then, for k = 2, 3, etc., blocks B(k) with the following features:

- 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,

- the block B(k) starts with the positive values 2^k+1, 2^k+2, ..., 2^(k+1),

- 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,

- then continues with the missing negative values down to -2^(k+1) + 1 with steps of -1 or -2.

As a consequence, nonnegative values appear in natural order.

Examples

			The first terms are:
  n   a(n)  |a(n)-a(n-1)|
  --  ----  -------------
   0     0  N/A
   1     1  2^0
   2    -1  2^1
   3    -2  2^0
   4     2  2^2
   5     3  2^0
   6     4  2^0
   7    -4  2^3
   8    -3  2^0
   9     5  2^3
  10     6  2^0
  11     7  2^0
  12     8  2^0
  13    -8  2^4
  14    -6  2^1
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..10000
• Rémy Sigrist, PARI program

Crossrefs

Cf. A377091, A377092.

Programs

PARI

\\ See Links section.