# What is the logic in this number sequence?

I’ve got a calendar with usually fairly simple logic or math puzzles, but was unable to solve this. You are allowed to use multiplication, sum, minus and division to calculate the next number.

The answer was 58, but there was no explaining formula, does anyone have a clue?

Update: an example of another sequence that was much easier to solve is this: 9 - 17 - 14 - 22 - 19 - ? Where the solution was simply alternating +8 and -3, so the answer here was 27.

• You're expected to add 3, then 9, then 18, then 27; but why 27, I don't know. Those are all suspiciously three-ful, though. – Patrick Stevens Feb 11 '18 at 9:35
• oeis.org/… has 12 sequences with $1,4,13,31$, but not one of them continues with 58. – Gerry Myerson Feb 11 '18 at 9:38
• Not very positive reviews of the calendar bol.com/nl/p/2018-neurocampus-braintraining-scheurkalender/… (doesn't necesarilly mean anything, but thought it is interesting) – Sil Feb 11 '18 at 9:55
• Maybe it has something to do with Dutch language? – Arnaud Mortier Feb 11 '18 at 10:23
• Normally this is just a math puzzle. I didn't found tricked questions yet on the calendar, and those sequences were pretty easy before. That's why I was so surprised and wondering if I missed something obvious. – Justus Romijn Feb 11 '18 at 10:34

## 2 Answers

With $$a_n=1+\frac{n(n+1)(n-1)}2,$$ the next term would be 61 instead.

Seriously, number sequences are highly arbitrary. Who would guess that the next number after $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59$ is $60$ - because it lists the orders of non-trivial simple groups?

• @quasi well it means that you can create a formula using those operands. So something like (N * 3 + 1) which works for 1,4,13 but doesn't for 31. Those sequences take usually the current number as input for a formula to calculate the next one. – Justus Romijn Feb 11 '18 at 10:33
• @hagen-von-eitzen I agree that number sequences can be super arbitrary, but in this case it involves a calendar that usually has fairly straightforward puzzles, so I was surprised that I could not figure out the formula behind this one. – Justus Romijn Feb 11 '18 at 10:40
• @hagen-von-eitzen: Apparently I misinterpreted the problem. – quasi Feb 11 '18 at 13:52

Got the answer from the creators of the calendar:

(translated from Dutch)

The number sequence is 1, 4, 13, 31, 58. You get to this in the following way:

Between 1 and 4 is 3, between 4 and 13 is 9, between 13 en 31 is 18, between 31 and 58 27.

This related to the table of 3's in this way:

1 X 3 = 3
3 X 3 = 9
6 X 3 = 18
9 X 3 = 27

This how you get to the next number.

The first step being 1 x 3 really messes up the sequence in my opinion, but yeah I guess that is just the problem with arbitrary number sequences.