A330261 Start with an empty stack S; for n = 1, 2, 3, ..., interpret the binary representation of n from left to right as follows: in case of bit 1, push the number 1 on top of S, in case of bit 0, replace the two numbers on top of S, say u on top of v, with u-v; a(n) gives the number on top of S after processing n.

1, 0, 1, -1, 1, 0, 1, -2, 1, 1, 1, -1, 1, 0, 1, -3, 1, 4, 1, 0, 1, 0, 1, -2, 1, 1, 1, -1, 1, 0, 1, -4, 1, 5, 1, 7, 1, 0, 1, -1, 1, 0, 1, 0, 1, 0, 1, -3, 1, 2, 1, 0, 1, 0, 1, -2, 1, 1, 1, -1, 1, 0, 1, -5, 1, 5, 1, -4, 1, 0, 1, 3, 1, -3, 1, 1, 1, 0, 1, -2, 1, -2

Offset: 1

Author

Rémy Sigrist, Dec 07 2019

Keywords

• sign
• base

Comments

This sequence is a variant of A308551.

After processing n, S has A268289(n) elements.

Every integer appears infinitely many times in the sequence:

- the effect of the binary string b(0) = "110" is to leave 0 on top of S,

- the effect of the binary string b(1) = "1" is to leave 1 on top of S,

- the effect of the binary string b(-1) = "11100" is to leave -1 on top of S,

- let "|" denote the binary concatenation,

- for any k > 0:

- the effect of b(k+1) = b(-1)|b(k)|"0" is to leave k+1 on top of S,

- the effect of b(-k-1) = b(1)|b(-k)|"0" is to leave -k-1 on top of S,

- for any k, for any n > 0, if the binary representation of n ends with b(k), then a(n) = k, QED,

- see A330264 for the values in order of appearance.

Examples

			The first terms, alongside the binary representation of n and the evolution of stack S, are:
  n   a(n)  bin(n)  S
  --  ----  ------  ------------------------------------------------------------
   1     1       1  () -> (1)
   2     0      10  (1) -> (1,1) -> (0)
   3     1      11  (0) -> (0,1) -> (0,1,1)
   4    -1     100  (0,1,1) -> (0,1,1,1) -> (0,1,0) -> (0,-1)
   5     1     101  (0,-1) -> (0,-1,1) -> (0,2) -> (0,2,1)
   6     0     110  (0,2,1) -> (0,2,1,1) -> (0,2,1,1,1) -> (0,2,1,0)
   7     1     111  (0,2,1,0) -> (0,2,1,0,1) -> (0,2,1,0,1,1) -> (0,2,1,0,1,1,1)
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..8192
• Rémy Sigrist, Scatterplot of the first 2^20 terms
• Rémy Sigrist, PARI program for A330261

Crossrefs

Cf. A268289, A308551, A330261, A330264.

Programs

PARI

See Links section.

Formula

a(2*k-1) = 1 for any k > 0.