# Generating ordered combination of numbers [closed]

I can form numbers with only 0,2,4,6,8.

The sequence is as follows 0,2,4,6,8,20,22,24,26,28,40,42,.....

How to generate an ordered sequence of numbers from combinations of 0,2,4,6,8.

• Write $N-1$ in base 5. Then double each digit. [I am using $N-1$ on the basis that 0 is the first member of the sequence.] – almagest Jun 8 '16 at 13:15
• Any number can be the nth number, lagrange extrapolation says so. – jimjim Jun 8 '16 at 13:27
• @Arjang Why should we use Lagrange extrapolation when there is deterministic algo to find the nth number. Please check the answer in the above comment and also in the below given answer. A big thanks to both of them to suggest the answer – Sayan Ghosh Jun 8 '16 at 13:44
• @SayanGhosh : deterministic algo? there are many algorithms to make up any number as the nth number and keep as many numbers as you define. This types of questions have been around for very long time, mathematically any rule that one can come up with is just one of infinity rules that seem to work. They are really good questions to make poeple think they have solved it while the any answer is as meaningless as the question. These are waste of time questions, there is nothing mathematical about making a rule that seems right. – jimjim Jun 8 '16 at 16:34
• meta.math.stackexchange.com/questions/21093/… – jimjim Jun 8 '16 at 16:38

Divide each number by 2 and then imagine that the resulting digit patterns are in base 5 rather than base 10. What are the numbers then?

Sequence:

$0, 2, 4, 6 ,8, 20, 22, 24, ...$

Divide by 2:

$0, 1, 2, 3, 4, 10, 11, 12, ...$

Treat digital patterns as if they were base 5:

$0_5, 1_5, 2_5, 3_5, 4_5, 10_5, 11_5, 12_5, ...$

Translate base 5 numbers back to base 10:

$0, 1, 2, 3, 4, 5, 6, 7, ...$

Hmmm...

To find the 100th number in the sequence, work this pattern backwards:

$100-1=99$

$99=344_5$

$344_{10}×2=688$

This should just about do it.

• What about just using any extrapolation formulae? nth number can be anything, 1, 0, 3.14, there are infinity many rules that work, what is special about this one rule? – jimjim Jun 8 '16 at 16:36
• We do not have an extrapolation of a finite number of terms! We have a specification for all terms, and the specification is specific enough to pick off any term without direct reference to the others. The listed initial terms are for illustration only. – Oscar Lanzi Jun 8 '16 at 19:58
• now that is a good point, yes this is creative work, not just another find the next number, apologies +1 :) – jimjim Jun 9 '16 at 6:06