24 May '15 19:337 edits

I made an extremely complex iterative numerical approach to give estimates of the output of an extremely complex mathematical function (to do with I new type probability that I invented and I am currently researching; but far to complex to explain here ) written in java computer language and I repeatedly run the program.

Because of the way the necessary equations had to be configured, although the input to this function is always a positive whole number, I was sure that the output would always be at a relatively arbitrary point somewhere along real number line and the chances of any being a whole number would be vertically zero. I was also not expecting to see any meaningful mathematical pattern in the output.

BUT, as I run the program again and again for different outputs, and, at a cost of each program run taking longer and longer (up to 20 minutes! ), as I made it iterate ever more times to get a more and more accurate output, I was astonished to noticed a pattern! As I made it more and more accurate, the numbers outputted seen to tend towards certain definite whole numbers, and so much so that I am certain it could not possibly be mere coincidence! This demands an explanation! But first I have to see if I can relate the output to the input algebraically so I can begin to do that. But that is where I have got stuck! Because I fail to see what algebraic equation could define how the output, which I call n, relates to the input, which I call v.

The table below shows each input v value on the left hand column and the corresponding output n on the right hand column.

OK, I hope someone smarter than me can spot what this mathematical patten is:

v (input ) | n (output )

0 --> 1 (note that this output of n=1 for v=0 is a special case and may or may not be relevant here)

1 --> 1

2 --> 3

3 --> 10

4 --> 35

5 --> 126

6 --> 462

7 --> 1716

8 --> 6435

9 --> 24310

10 --> 92378 ?

11 --> 352716 OR 352715 ?

12 --> 1352079 OR 1352078 ?

Some important notes:

Firstly, the input of v=0 there is a SPECIAL CASE and therefore may or may not be mathematically consistent with the outputs given v>0. So, if you think you have almost spotted the pattern but not quite because your equation doesn't work for v=0 above, you are permitted to ignore the special case for v=0 and you have got the answer.

Secondly, note the question marks indicating some uncertainly of the output above. The iteration in my java program works well for the first ~7 or so inputs i.e. v < ~8, but then, as you go down the above table, it gets more and more uncertain which whole number the output tends towards, although the figures above cannot be too far out. So, if you think you have almost spotted the pattern but not quite because your equation doesn't work for v > ~7 above, you are permitted to slightly adjust, but only slightly adjust, those output values for inputs greater than about ~7 to make them fit with your equation.

Obviously, what I ideally want is some equation in the form of:

n = ( ...some expression here with v in it... )

I will be very grateful to anyone's help here.

Because of the way the necessary equations had to be configured, although the input to this function is always a positive whole number, I was sure that the output would always be at a relatively arbitrary point somewhere along real number line and the chances of any being a whole number would be vertically zero. I was also not expecting to see any meaningful mathematical pattern in the output.

BUT, as I run the program again and again for different outputs, and, at a cost of each program run taking longer and longer (up to 20 minutes! ), as I made it iterate ever more times to get a more and more accurate output, I was astonished to noticed a pattern! As I made it more and more accurate, the numbers outputted seen to tend towards certain definite whole numbers, and so much so that I am certain it could not possibly be mere coincidence! This demands an explanation! But first I have to see if I can relate the output to the input algebraically so I can begin to do that. But that is where I have got stuck! Because I fail to see what algebraic equation could define how the output, which I call n, relates to the input, which I call v.

The table below shows each input v value on the left hand column and the corresponding output n on the right hand column.

OK, I hope someone smarter than me can spot what this mathematical patten is:

v (input ) | n (output )

0 --> 1 (note that this output of n=1 for v=0 is a special case and may or may not be relevant here)

1 --> 1

2 --> 3

3 --> 10

4 --> 35

5 --> 126

6 --> 462

7 --> 1716

8 --> 6435

9 --> 24310

10 --> 92378 ?

11 --> 352716 OR 352715 ?

12 --> 1352079 OR 1352078 ?

Some important notes:

Firstly, the input of v=0 there is a SPECIAL CASE and therefore may or may not be mathematically consistent with the outputs given v>0. So, if you think you have almost spotted the pattern but not quite because your equation doesn't work for v=0 above, you are permitted to ignore the special case for v=0 and you have got the answer.

Secondly, note the question marks indicating some uncertainly of the output above. The iteration in my java program works well for the first ~7 or so inputs i.e. v < ~8, but then, as you go down the above table, it gets more and more uncertain which whole number the output tends towards, although the figures above cannot be too far out. So, if you think you have almost spotted the pattern but not quite because your equation doesn't work for v > ~7 above, you are permitted to slightly adjust, but only slightly adjust, those output values for inputs greater than about ~7 to make them fit with your equation.

Obviously, what I ideally want is some equation in the form of:

n = ( ...some expression here with v in it... )

I will be very grateful to anyone's help here.