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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. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,8

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.

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

FORMULA

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

EXAMPLE

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)

PROG

(PARI) See Links section.

CROSSREFS

Cf. A268289, A308551, A330261, A330264.

Sequence in context: A206706 A302354 A000164 * A157746 A349082 A281010

Adjacent sequences:  A330258 A330259 A330260 * A330262 A330263 A330264

KEYWORD

sign,base

AUTHOR

Rémy Sigrist, Dec 07 2019

STATUS

approved

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Last modified January 26 12:39 EST 2022. Contains 350598 sequences. (Running on oeis4.)