Mathematical Puzzle

[Alan]

Given the first few terms of y = f(x), can you determine what the function f() is?

x = 1, 2, 3, 4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20
y = 0, 1, 1, 0, -1, -1, -1,  0,  1,  2,  2,  2,  2,  1,  0, -1, -2, -2, -2, -2

I should point out that I don't know the answer, although I have written a generator which works perfectly, however I'm specifically looking for a single formula.

[a_bertrand]

As it goes up and down it seems to be something related to a sin or cos. As the numbers are also int, I would say there is a rounding too as well as like a second sin to increase the effect over time. Remains to tweak all those to find the right effect. Other option use Mathlab and it's curve fitting to let it find it for you ;)

[sniko]

Ah, I thought I was on to something, until I got to x=5.

1/sqrt(1+0^0)
2/sqrt(2+1^1)
3/sqrt(3+2^2)
4/sqrt(4+3^3)

x / sqrt(x+(x-1)^(x-1))

[dnenb]

Do you know anything about the function? Does it take and produce only integers? As bertrand said you could easily use matlab or an equivalent package to fit a function to the data:

[ATTACH=CONFIG]1233[/ATTACH]

[a_bertrand]

here is an approximation (a few values are not appearing at the right time but you can continue to tweak the formula):

a=1.1

b=0.08

c=0.46

d=1

e=0.28

y=round(sin((x-c)/a+(e*x)))*(x*b)+d))

[ATTACH=CONFIG]1234[/ATTACH]

Series 2 (and column C) are your original values, while column B and series 1 is my current approximation. Mathlab or excel can both help you ;)

[Alan]

Some interesting suggestions, I hadn't considered trig functions.

@denb; yes only integers are produced.

For anyone who is curious:

<?php

$index     = 1;
$x         = 0;
$y         = 0;
$direction = 0;
$matrix     = array(array(1, 0), array(0, 1), array(-1, 0), array(0, -1));

for ($i = 0; $i < 10; $i++) /* increment 10 to suit */
{
   for ($j = 0; $j < floor(($i + 2) / 2); $j++)
   {
       echo sprintf("%d : ( %d , %d )\n", $index, $x, $y);

       $index++;

       $x += $matrix[$direction][0];
       $y += $matrix[$direction][1];
   }

   $direction = ++$direction % 4;
}

If you plot the generated coordinates; you end up with a rather remarkable and handy ""curve"".