The Online Encyclopedia of Integer Sequences contains over 200,000 sequences. It contains classics, curios, thousands of derivatives entered purely for completeness’s sake, short sequences whose completion would be a huge mathematical achievement, and some entries which are just downright silly.

For a lark, David and I have decided to review some of the Encyclopedia’s sequences. We’ll be rating sequences on four axes: Novelty, Aesthetics, Explicability and Completeness.

A010727
Constant sequence: the all 7’s sequence.

7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, ...

Novelty

David: I can already think of a similar sequence for every other natural number.

Christian: But seven, though. That’s quite special. I applaud the author’s bold vision.

David: Almost no integer sequences are constant.

Christian: So it’s basically unique?

David: Yes.

\[ \frac{5}{5} \]

Aesthetics

David: Beautiful. It’s sexy and it knows it.

Christian: Wouldn’t that be hepty?

David: I’m in seventh heaven looking at this.

Christian: Coin in the pun jar.

David: It was worth it.

Christian: Score?

\[ \frac{5}{5} \]

Explicability

David: Simple. I think I’ve found a closed formula.

Christian: Go on then.

David: \[ F_{n+8} = \frac{\sum_{i=1}^7 F_{n+i}}{7} \]

(with initial conditions $F_0 = F_1 = F_2 = F_3 = F_4 = F_5 = F_6 = 7$)

Christian: That’s a terrible explanation.

David: We should try some different initial conditions.

Christian: No.

\[ \frac{5}{5} \]

Completeness

Christian: This one’s a gimme. It’s a constant sequence. How many entries does the OEIS have?

David: 80. We should submit the 81st. According to my formula, it’s…

…

David: \[(7+7+7+7+7+7+7) \div 7 = 7.\] It’s $7$.

Christian: Maybe asking for the whole sequence is a bit much, but I want more than 80. I reckon a low score is due. I suggest A010716 as a decimal.

\[ \frac{0.55555555555…}{5} \]

Final score

\[ \frac{5+5+5+0.\dot{5}}{20} = \frac{7}{9} = 0.77777777…. \]

Christian: Poetic.