This article is about the Linus Sequence. As far as I know, it’s the only number sequence that is named after a cartoon strip character (This being the internet, I’m sure I’ll rapidly learn if there are others! I’ll update this article if people contact me with other named sequences). The Linus Sequence (OEIS: A006345) is named after the character, Linus van Pelt, from the popular Peanuts Cartoon. It’s in reference/homage to the cartoon strip see below: |

The sequence was first described by Nathaniel Hellerstein, in 1978. It’s a sequence that, ironically, is extended by trying to make a “pattern-breaking pattern”.

The Linus sequence is a sequence of “Trues” and “Falses” in which each new entry to the end of the sequence is chosen to prevent the longest possible pattern from emerging at the end of the sequence.

Here are the first few entries:

**T F T T F F T F T T F T F F T T F T T T **…

Admittedly, this isn’t exactly the sequence that Linus was describing in the comic strip, but let’s take a closer look about how the sequence grows, and how, mathematically we make a “pattern-breaking pattern”.

The sequence, by definition, starts with

**T**.

At each stage, there is a choice to add either a **T** or **F** to the end of the current sequence.

Which character is chosen, is the character that *minimizes* the length of any immediately repeated sequence that now occurs at the end. It’s a fun little exercise to knock out code to generate the sequence (it’s popular with code golfers).

It’s easy to see through example.

After the first **T**, if we added another **T** to the end, we’d get the repeated sequence of **TT**, which is two units long. Conversely, if we added a **F** to the end there is no repeat, so adding the **F** minimizes the repeating pattern at the end of the string.

We are now at

**TF**.

If we added another **F** to the end, we’d get the repeated string **FF** at the end, so instead we revert to **T**.

We are now at

**TFT**.

We have to now add another **T**, otherwise, we’d end up with **TFTF**, which is double repeat of **TF**.

We are now at

**TFTT**.

Adding another **T** to this string would result in **TT** at the end (length two), so instead we add another **F**.

We are now at

**TFTTF**.

After this. We can’t add another **T** as this would cause the repeated sequence **TFT** (the entire string would be **TFTTFT**), so we have the add **F**.

We are now at

**TFTTFF**.

And so on …

It seems a little confusing at first, but the simple rule is that you select the character that *avoids* the longest possible “double suffix” forming at the end of the sequence.

Here are the first 500 terms of the sequence:

TFTTFFTFTTFTFFTTFTTTFFTFTTFFTTTFTTFFTFTTFTFFTTFTTTFFTFTTFFTFFFTTFTFFTTFFFTFFTTFTFFTFTTFFTFFFTTFTFFTTFFFTFFTTTFTTFFTFTTFTFFTTFTTTFFTFTTFFTTTFTTFFTFTTFTFFTTFTTTFFTFFFTTFTFFTTFFFTFFTTFTFFTFTTFFTFFFTTFTFFTTFFFTFFTTFTFFTTTFTTFFTFTTFTFFTTFTTTFFTFTTFFTTTFTTFFTFTTFTFFTTFTTTFFTFFFTTFTFFTTFFFTFFTTFTFFTFTTFFTFFFTTFTFFTTFFFTFFTFTTFFTFFFTTFTFFTTFTTTFFTFTTFFTTTFTTFFTFTTFTFFTTFTTTFFTFTTFFTTTFTTFFFTFFTTFTFFTFTTFFTFFFTTFTFFTTFFFTFFTTFTFFTFTTFFTFFFTTFTTTFFTFTTFFTTTFTTFFTFTTFTFFTTFTTTFFTFTTFFTTTFTTFFTFTTFFFTFFTTFT

It's more common to see this sequence expressed as numbers *e.g.* 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1 …

A closely related sequence, called the Sally Sequence, actually makes the Linus sequence even easier to understand. The Sally sequence donates the length of repetition that is It is named after Sally, Charlie Brown’s little sister. Sally has a crush on Linus (her “Sweet Babboo”). |

This is what the first few terms of the Sally Sequence look like:

0, 1, 1, 2, 1, 3, 1, 1, 3, 2, 1, 6, 3, 2, 1, 3, 1, 1, 6, 3, 2, 4, 1, 1, 3, 2, 1, 3, 1, 6, 4, 2, 1, 2, 4, 3, 1, 8 …

Interestingly, the values in the Sally Sequence appear quantized, in the first 1,000 terms, you only find the values: 0, 1, 2, 3, 4, 6, 8, 24, 30, 108 in the sequence.

If increase the window to 20,000 terms still the only values seen are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 16, 18, 24, 30, 108, 528, 552, 1298, 2752, 5876.

If you calculate the sequent to 1,000,000 terms, the only vales seen are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 24, 30, 36, 60, 72, 108, 318, 372, 414, 420, 444, 522, 528, 546, 552, 1280, 1292, 1298, 1632, 2620, 2752, 5876, 6203, 6218, 13912, 14312, 17220, 17580, 31532, 87650.

If we take **T** to be +1, and **F** to be -1, we can build a cumulative running total of the sequence like a drunken-man's walk. It's pretty easy to see the structure in this sequence from the this graph. Here is graphed the first 800 terms.

Similarly, here is a plot of the first 800 terms of the Sally Sequence.

For more information on the Peanuts comic strips, cartoons, characters, videos, shops, and museum, please visit the official Peanuts web page at peanuts.com

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