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Archive Labeling Sequences

Tanya Khovanova, Gregory Marton

Abstract

What follows is the story of a family of integer sequences, which started life as a Google interview puzzle back in the previous century when VHS video tapes were in use.

Archive Labeling Sequences

Abstract

What follows is the story of a family of integer sequences, which started life as a Google interview puzzle back in the previous century when VHS video tapes were in use.
Paper Structure (11 sections, 8 theorems, 20 equations, 1 figure, 6 tables)

This paper contains 11 sections, 8 theorems, 20 equations, 1 figure, 6 tables.

Table of Contents

  1. Google's Puzzle
  2. Ones Counting Function
  3. Counting Other Digits
  4. More "Exactly" Sequences
  5. Counting Zeroes
  6. Greater or Equal
  7. The Algorithms
  8. Multiple Digits
  9. All Your Base
  10. Future research
  11. Acknowledgments

Key Result

Lemma 5.1

For any integer $x > 10^{10}$, we have $f_0(x+10^{10}) \ge f_0(x) + 10^{10}$.

Figures (1)

  • Figure 1: One of the sticker sheets that came with early VHS tapes, with gratitude to an r/nostalgia user vhs-image.

Theorems & Definitions (19)

  • Definition 3.1
  • Definition 3.2
  • Definition 4.1
  • Lemma 5.1
  • proof
  • Theorem 5.2
  • proof
  • Lemma 7.1
  • proof
  • Proposition 8.1
  • ...and 9 more