A347411 Sqrt(3)+1-adic expansion of 4, in balanced ternary alphabet.

0, 0, 0, 0, 1, 0, 1, -1, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0, 1, 0, 1, -1, 1, -1, 0, -1, 0, -1, 1, -1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, -1, 1, 0, 0, -1, 0, 0, 0, 0, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 0, 1, 0, 1, 0, 0, -1, 1, -1, 1, -1, 1, -1, 0, -1, 1, -1, 0, -1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, -1, 1, 0, 1, 0, 0, -1, 1, -1, 0, 0, 0, 0, 0, 0, 1, -1, 1, -1, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, -1, 0, 0, 0, -1, 1, 0, 1, 0, 1, 0, 0, 0, 0, -1, 0, 0, 1, -1, 1, -1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 1, -1, 1, -1, 1, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 1, -1, 0, -1, 0, 0, 1, 0, 0, 0, 0, -1, 1, 0

Offset

0

Comments

Starting with the number 4, repeatedly "carry" into higher terms of the sequence by scaling and shifting the sequence (-2, -2, 1). For odd numbers, retain sign by scaling the carry by half its magnitude, rounded toward 0. See examples for iterates.

Also appears to work with initial sequences [2, 0, 1, -1, 1, -1, ...] (repeating 1 and -1), [0, 0, 2, -2, 2, -2] (repeating 2 and -2), and [0, 0, 0, 0, 1, -2, 3, -4] (alternating naturals).

Sequence name is by analogy with p-adic numbers and due to the carry sequence (-2, -2, 1) corresponding to the polynomial x^2 - 2x - 2, which has root sqrt(3)+1.

1 only occurs at even positions, -1 occurs only at odd positions.

First 8192 digits appear to be aperiodic.

Links

Table of n, a(n) for n=0..199.

Ben Conner, Polynomial Counting: Two 2’s

Example

First few iterates of carry, starting from expansion "4":
  4;
  0,  -4,   2;
  0,   0,   6,  -2;
  0,   0,   0,  -8,   3;
  0,   0,   0,   0,  11,   4;
  0,   0,   0,   0,   1, -14,   5;
  0,   0,   0,   0,   1,   0,  19,  -7;
  0,   0,   0,   0,   1,   0,   1, -25,   9;
  0,   0,   0,   0,   1,   0,   1,  -1,  33, -12;
In the limit, this produces the sequence above.

Prog

(Haskell)
a347411_iterates = iterate (doubleBorrow []) (4:repeat 0) where
  doubleBorrow zs (x:y:z:xs)
    | abs x >= 2 = let (q, r) = quotRem x 2 in foldl (flip (:)) (r : y-x+r : z+q : xs) zs
    | otherwise  = doubleBorrow (x:zs) (y:z:xs)
a347411 n = (a347411_iterates !! (n+1)) !! n

Crossrefs

Sequence in context: A333189 A011662 A011665 * A059437 A152592 A347412

Adjacent sequences: A347408 A347409 A347410 * A347412 A347413 A347414

Keyword

sign,base

Author

Ben Conner, Aug 30 2021

Status

approved