# What comes next in the sequence?

My roommate has insomnia so he was counting sheep to help him sleep.

He started with $$2986$$, then continued with $$47786,\ 764586,$$ and $$12233386$$.

What sequence is he following, and what was the next number?

• We see a champion rising; congrats on becoming the #1 puzzler of the month! – Ébe Isaac Mar 15 '20 at 3:56

The next number in the sequence is:

195 734 186

The sequence he is following is:

BAA = 2 986
BAAA = 47 786
BAAAA = 764 586
BAAAAA = 12 233 386
BAAAAAA = 195 734 186

• Yep, you got it. – DenverCoder1 Mar 11 '20 at 19:57
• ... black sheep? – Rand al'Thor Mar 12 '20 at 14:20
• I didn't think of it. It's so obvious now when I have to count sheep as well. – tansy Mar 13 '20 at 1:32
• Normally, I don't like sequence puzzles like this, but this answer is great. – Kevin Mar 13 '20 at 17:02

I found it's

195734186

It's sequence where

k(n+1) = k(n) * 16 + 10

• your solution is equivalent to the others :) – daw Mar 13 '20 at 12:21
• It's true, it's also "less right" because of "counting sheep" hint but is also different than "right one", it shows that there can be different way to solve the same problem. – tansy Mar 14 '20 at 21:31

The sequence is: $$k(n+1) = 16 \cdot k(n) + 10$$ for the initial condition $$k(1) = 2986$$.

This recurrence equation can be solved as follows: $$k(n) = \frac{70 \cdot 2^{4 n + 3}}{3} - \frac{2}{3}$$ This formula can be used to calculate the $$n^{th}$$ value in this sequence for any value of $$n$$.

$$k(1) = 2986$$ $$k(2) = 47786$$ $$k(3) = 764586$$ $$k(4) = 12233386$$ $$k(5) = 195734186$$ $$\dots$$
The pattern of multiplying by 16 and adding 10 is the same as (in binary) performing a right-shift operation 4 times and then adding 1010. In hexadecimal, 4 bits is one digit and 1010 = A. In other words, $$16\cdot k(n)+10\$$ takes the representation of $$k(n)$$ in hexadecimal and appends $$\text{A}$$.

$$2986$$ = $$\text{BAA}$$.
$$47786 = \text{BAAA}$$.
$$764586 = \text{BAAAA}$$.
$$12233386 = \text{BAAAAA}$$.
$$195734186 = \text{BAAAAAA}$$.

It makes sense that he would be using this sequence to count sheep since $$\text{BAA}\dots$$ is the sound the sheep make.

• While this is correct, and thorough, it misses the funny part, which is that expressed in hexadecimal these spell (rot13) "ONN", "ONNN", ONNNN" ... fbhaqf gung furrc znxr -- naq pbhagvat furrc vf n genqvgvbany npgvivgl sbe bpphclvat lbhe zvaq gb uryc lbh snyy nfyrrc. – Ross Presser Mar 13 '20 at 17:49