# Restore a once very happy sequence

Once, I had a nice sequence of numbers.. They were all very happy.

Suddenly, someone came by and stole all the lines. The numbers didn't know what to do, and got very scared. They clung to each other.

It's up to you to restore the sequence. Also, what are the next numbers? And why?

1616131256113216312...

hint:

the lines stolen may be fraction bars.

• Good grief the numbers are bolded! Commented Jul 7, 2023 at 12:42
• Is it relevant at all that the numbers 0, 4, 7, 8, and 9 do not appear in this sequence? Or that 5 only appears once? Commented Jul 13, 2023 at 16:06
• @newQOpenWid, not much. in small numbers they will not be there very often, however in (very) large numbers they are going to appear. Commented Jul 13, 2023 at 18:35
• Just wanted to toss this in: is 'happy' a reference to happy numbers or am I just too eager to make connections? Commented Jul 22, 2023 at 22:00
• @newQOpenWid, no, its just the story. Commented Jul 22, 2023 at 22:11

This sequence is an obfuscation of a classic mathematical sequence...

The Fibonacci sequence, which begins $$1$$, $$1$$, $$2$$, $$3$$, $$5$$, $$8$$, $$13$$, $$21$$, $$34$$, ... with each term being the sum of the previous two.

The numbers in this string have been obtained by...

...taking each term of the Fibonacci sequence, dividing it by 6, cancelling down the numerator and denominator, presenting it as a 'mixed number' (i.e. the greatest whole number possible, with the remainder as a fraction), then concatenating all digits and removing the fraction bars.

This way...

$$1$$, $$1$$, $$2$$, $$3$$, $$5$$, $$8$$, $$13$$, $$21$$, ...

on dividing by 6, becomes:

$$\frac16$$, $$\frac16$$, $$\frac26$$, $$\frac36$$, $$\frac56$$, $$\frac86$$, $$\frac{13}6$$, $$\frac{21}6$$, ...

which, cancelling down the fractions, becomes:

$$\frac16$$, $$\frac16$$, $$\frac13$$, $$\frac12$$, $$\frac56$$, $$\frac43$$, $$\frac{13}6$$, $$\frac72$$, ...

which, presented as mixed numbers, becomes:

$$\frac16$$, $$\frac16$$, $$\frac13$$, $$\frac12$$, $$\frac56$$, $$1\frac13$$, $$2\frac16$$, $$3\frac12$$, ...

which, concatenated and with fraction bars removed, becomes:

1616131256113216312...

Here you can see that the numbers in bold in the string represent the whole number parts of the mixed numbers.

The next few digits in the string must therefore be derived by applying the same transformation to...

...the next term of the Fibonacci sequence: 34.

Since $$\frac{34}6$$ is $$5\frac23$$, these next digits will be 523.