TLDR; I go on a math adventure and get overwhelmed :)
My maths isn't great (I can't read notation) but I'm a competent programmer and reasonable problem solver. I've done the first dozen or so Euler problems and intend to continue with that when I have time.
In Arthur C Clarke's story "The 9 Billion Names of God" the names of God are all possible sequences in an unspecified alphabet, having no more than nine characters, where no letter occurs more than three times in succession.
Out of curiosity, I started playing around with determining how many valid sequences there are in a range.
I started with digits repeating in base 10 numbers, at heart it's the same problem as letters repeating in an alphabet.
Not being very knowledgable about math, I thought I'd write a program iterate over ranges and count all the elements that match the above condition, then put the results in a spreadsheet to see if a clear pattern of some kind emerged that would let me write an algorithm to determine the number of valid sequences in a given range.
I started with the constraint that a digit could only appear once, so in the range 0-99 there are 9 invalid sequences, 11, 22, 33 etc., leaving 91 valid 'names of God'.
Here's the table for 0-99 through 0-99999999. I stopped there because beyond that's where it started taking to long to calculate and I didn't want to get sidetracked optimizing.
0-99 91 0-999 820 0-9999 7381 0-99999 66430 0-999999 597871 0-9999999 5380840 0-99999999 48427561
I also generated a table for digits appearing no more than twice or thrice:
0-999 991 0-9999 9820 0-99999 97300 0-999999 964081 0-9999999 9552430 0-99999999 94648600 0-9999 9991 0-99999 99820 0-999999 997300 0-9999999 9964000 0-99999999 99550081
I haven't got around to looking into these yet, because I got fascinated by the first table.
The first table appears in OEIS as A002452.
Going from there, looking at all sorts of different things, amongst them the sequences of numbers in different placeholder columns in the tables, differences between numbers in different columns and/or tables etc. I looked at all sorts of things, I wish I'd documented it more, I was just idly mucking around with a spreadsheet and Googling sequences. With a quick Google search I found some of these sequences in all sorts of strange places, some examples include transformations of Lucas Numbers, solutions to Kakuro / Addoku / Soduku puzzles, repunits, the coordinates of geodesic faces, even the Ishango bone, which I'd never heard of before. It justs goes on and on.
Maths is full of this sort of thing isn't it? And I'm just looking at one little problem from one very specific angle, this is just the tip of the iceberg here isn't it?
Questions/requests for comments:
I'm presuming that my adventure isn't anything extraordinary at all and maths is full of this unexpected relationships stuff, true?
What is the right way to describe the problem outlined in the first few paragraphs, what do I need to learn to figure it out?
I'd love to hear any comments/trivia etc. relating to this little adventure please!