Finding a formula for a pattern

Question

I have this pattern which is an infinite sequence (I have placed commas so it's easy to see the pattern)...

$1 ,1 2, 1 2 3, 1 2 3 4, 1 2 3 4 5 ...$

Is there any formula governing this sequence, ie, If I gave you an index, would you be able to run it through a formula that outputs the digit at that index?

Answer 1

The sequence $$1,2,3,4,5,6,7,8,9,1,0,1,1,1,2,1,3,1,4\dots$$ which is what the given sequence is "converging to" is listed at the Online Encyclopedia of Integer Sequences. A formula for the $n$th term, due to David Cantrell, is given there, but you'll have to look up the Lambert W-function to understand it.

Let "index" $i = ceiling( W(\log(10)/10^{1/9} (n - 1/9))/\log(10) + 1/9 )$ where $W$ denotes the principal branch of the Lambert $W$ function. Then $$a(n) = mod(floor(10^{mod(n + (10^i - 10)/9, i) - i + 1} ceiling((9n + 10^i - 1)/(9i) - 1)), 10)$$

Some of that is a bit confusing, but aside from doing a little bit of TeX formatting I have tried to copy it as it's given at the OEIS.

Answer 2

Not a complete answer to your question, but it reminds me of the factorial number system.

Under this mixed-radix system, the sequence $A_n=(n!-1)$ yields:

• $0$
• $10$
• $210$
• $3210$
• $43210$
• $543210$
• $\dots$