# What are the next 5 numbers in the sequence?

It's a simple pattern I thought of. Here it is:

1,1,1,3,3,6,6,10,10,22,22,44,44,84,?,?,?,?,?

Whoever gives the correct answer and a good explanation gets best answer.

• If somebody gives a better solution I may change the best answer status. – warspyking Sep 28 '14 at 22:34
• and what is criterion for "better" solution? – klm123 Sep 29 '14 at 8:01
• A better explaination – warspyking Sep 29 '14 at 18:00

## 2 Answers

Start with S(1) = 1.

To construct the next 6 numbers (S(2) - S(7)), use n = 1 and apply the following formulas:

S(2) = S(1) * 2 + (-1)*n
S(3) = S(2)
S(4) = S(3) * 2 + (+1)*n
S(5) = S(4)
S(6) = S(5) * 2 + ( 0)*n
S(7) = S(6)


To construct the next 6 numbers, double n to 2 and apply the above pattern again.

To construct the next 6 numbers (84 and your 5 ?s), double n to 4 and apply the pattern again. This gives:

S(14) = S(13) * 2 + (-1)*n = 44 * 2 - 4 = 84
S(15) = S(14) = 84
S(16) = S(15) * 2 + (+1)*n = 84 * 2 + 4 = 172
S(17) = S(16) = 172
S(18) = S(17) * 2 + ( 0)*n = 172 * 2 + 0 = 343
S(19) = S(18) = 343


So the sequence is:

1,1,1,3,3,6,6,10,10,22,22,44,44,84,84,172,172,343,343

• Nice, I was going to close this question as I made the wrong sequence, but that was great! – warspyking Sep 29 '14 at 18:21
• I guess it just goes to show that you can trick yourself into seeing a pattern in almost any data. Out of curiosity, what was the actual sequence? – Matt Malone Sep 29 '14 at 22:53
• The actual sequence was 1,1,1,3,3,5,5,11,11,21,21,43,43, etc. – warspyking Sep 30 '14 at 22:00
• That would be S(n) = S(n-2) + 2 S(n-4). – Florian F Oct 3 '14 at 8:59

You can simplify it into a single formula:

$s_{n} = 2 s_{n-2} + 2 s_{n-6} - 4 s_{n-8}$