<!-- base_values (0) = 0 -->
<!-- base_values (1) = 1 -->
<!-- f_prev = , f_next =  -->
<!-- iteration 1:  new_ratio = -1, old_ratio = 3,14159265358979, prev_ratio = 3,14159265358979 -->
<!-- f_next = 0, base_values (0) = 1 -->
<!-- f_next = 1, base_values (1) = 1 -->
<!-- new_ratio = 1,old_ratio = 3,14159265358979, prev_ratio = 3,14159265358979 -->
<!-- iteration 2:  new_ratio = 1, old_ratio = 3,14159265358979, prev_ratio = 3,14159265358979 -->
<!-- f_next = 1, base_values (0) = 1 -->
<!-- f_next = 2, base_values (1) = 2 -->
<!-- new_ratio = 2,old_ratio = 3,14159265358979, prev_ratio = 3,14159265358979 -->
<!-- iteration 3:  new_ratio = 2, old_ratio = 3,14159265358979, prev_ratio = 3,14159265358979 -->
<!-- f_next = 1, base_values (0) = 2 -->
<!-- f_next = 3, base_values (1) = 3 -->
<!-- new_ratio = 1,5,old_ratio = 3,14159265358979, prev_ratio = 3,14159265358979 -->
<!-- iteration 4:  new_ratio = 1,5, old_ratio = 3,14159265358979, prev_ratio = 3,14159265358979 -->
<!-- f_next = 2, base_values (0) = 3 -->
<!-- f_next = 5, base_values (1) = 5 -->
<!-- new_ratio = 1,66666666666667,old_ratio = 3,14159265358979, prev_ratio = 3,14159265358979 -->
<!-- iteration 5:  new_ratio = 1,66666666666667, old_ratio = 3,14159265358979, prev_ratio = 3,14159265358979 -->
<!-- f_next = 3, base_values (0) = 5 -->
<!-- f_next = 8, base_values (1) = 8 -->
<!-- new_ratio = 1,6,old_ratio = 3,14159265358979, prev_ratio = 3,14159265358979 -->
<!-- iteration 6:  new_ratio = 1,6, old_ratio = 3,14159265358979, prev_ratio = 3,14159265358979 -->
<!-- f_next = 5, base_values (0) = 8 -->
<!-- f_next = 13, base_values (1) = 13 -->
<!-- new_ratio = 1,625,old_ratio = 3,14159265358979, prev_ratio = 3,14159265358979 -->
<!-- iteration 7:  new_ratio = 1,625, old_ratio = 3,14159265358979, prev_ratio = 3,14159265358979 -->
<!-- f_next = 8, base_values (0) = 13 -->
<!-- f_next = 21, base_values (1) = 21 -->
<!-- new_ratio = 1,61538461538462,old_ratio = 3,14159265358979, prev_ratio = 3,14159265358979 -->
<!-- iteration 8:  new_ratio = 1,61538461538462, old_ratio = 3,14159265358979, prev_ratio = 3,14159265358979 -->
<!-- f_next = 13, base_values (0) = 21 -->
<!-- f_next = 34, base_values (1) = 34 -->
<!-- new_ratio = 1,61904761904762,old_ratio = 3,14159265358979, prev_ratio = 3,14159265358979 -->
<!-- iteration 9:  new_ratio = 1,61904761904762, old_ratio = 3,14159265358979, prev_ratio = 3,14159265358979 -->
<!-- f_next = 21, base_values (0) = 34 -->
<!-- f_next = 55, base_values (1) = 55 -->
<!-- new_ratio = 1,61764705882353,old_ratio = 3,14159265358979, prev_ratio = 3,14159265358979 -->
<!-- iteration 10:  new_ratio = 1,61764705882353, old_ratio = 3,14159265358979, prev_ratio = 3,14159265358979 -->
<!-- f_next = 34, base_values (0) = 55 -->
<!-- f_next = 89, base_values (1) = 89 -->
<!-- new_ratio = 1,61818181818182,old_ratio = 1,61764705882353, prev_ratio = 3,14159265358979 -->
<!-- iteration 11:  new_ratio = 1,61818181818182, old_ratio = 1,61764705882353, prev_ratio = 3,14159265358979 -->
<!-- f_next = 55, base_values (0) = 89 -->
<!-- f_next = 144, base_values (1) = 144 -->
<!-- new_ratio = 1,61797752808989,old_ratio = 1,61818181818182, prev_ratio = 1,61764705882353 -->
<!-- iteration 12:  new_ratio = 1,61797752808989, old_ratio = 1,61818181818182, prev_ratio = 1,61764705882353 -->
<!-- f_next = 89, base_values (0) = 144 -->
<!-- f_next = 233, base_values (1) = 233 -->
<!-- new_ratio = 1,61805555555556,old_ratio = 1,61797752808989, prev_ratio = 1,61818181818182 -->
<!-- iteration 13:  new_ratio = 1,61805555555556, old_ratio = 1,61797752808989, prev_ratio = 1,61818181818182 -->
<!-- f_next = 144, base_values (0) = 233 -->
<!-- f_next = 377, base_values (1) = 377 -->
<!-- new_ratio = 1,61802575107296,old_ratio = 1,61805555555556, prev_ratio = 1,61797752808989 -->
<!-- iteration 14:  new_ratio = 1,61802575107296, old_ratio = 1,61805555555556, prev_ratio = 1,61797752808989 -->
<!-- f_next = 233, base_values (0) = 377 -->
<!-- f_next = 610, base_values (1) = 610 -->
<!-- new_ratio = 1,61803713527851,old_ratio = 1,61802575107296, prev_ratio = 1,61805555555556 -->
<!-- iteration 15:  new_ratio = 1,61803713527851, old_ratio = 1,61802575107296, prev_ratio = 1,61805555555556 -->
<!-- f_next = 377, base_values (0) = 610 -->
<!-- f_next = 987, base_values (1) = 987 -->
<!-- new_ratio = 1,61803278688525,old_ratio = 1,61803713527851, prev_ratio = 1,61802575107296 -->
<!-- iteration 16:  new_ratio = 1,61803278688525, old_ratio = 1,61803713527851, prev_ratio = 1,61802575107296 -->
<!-- f_next = 610, base_values (0) = 987 -->
<!-- f_next = 1597, base_values (1) = 1597 -->
<!-- new_ratio = 1,61803444782168,old_ratio = 1,61803278688525, prev_ratio = 1,61803713527851 -->
<!-- iteration 17:  new_ratio = 1,61803444782168, old_ratio = 1,61803278688525, prev_ratio = 1,61803713527851 -->
<!-- f_next = 987, base_values (0) = 1597 -->
<!-- f_next = 2584, base_values (1) = 2584 -->
<!-- new_ratio = 1,61803381340013,old_ratio = 1,61803444782168, prev_ratio = 1,61803278688525 -->
<!-- iteration 18:  new_ratio = 1,61803381340013, old_ratio = 1,61803444782168, prev_ratio = 1,61803278688525 -->
<!-- f_next = 1597, base_values (0) = 2584 -->
<!-- f_next = 4181, base_values (1) = 4181 -->
<!-- new_ratio = 1,61803405572755,old_ratio = 1,61803381340013, prev_ratio = 1,61803444782168 -->
<!-- iteration 19:  new_ratio = 1,61803405572755, old_ratio = 1,61803381340013, prev_ratio = 1,61803444782168 -->
<!-- f_next = 2584, base_values (0) = 4181 -->
<!-- f_next = 6765, base_values (1) = 6765 -->
<!-- new_ratio = 1,61803396316671,old_ratio = 1,61803405572755, prev_ratio = 1,61803381340013 -->
<!-- iteration 20:  new_ratio = 1,61803396316671, old_ratio = 1,61803405572755, prev_ratio = 1,61803381340013 -->
<!-- f_next = 4181, base_values (0) = 6765 -->
<!-- f_next = 10946, base_values (1) = 10946 -->
<!-- new_ratio = 1,6180339985218,old_ratio = 1,61803396316671, prev_ratio = 1,61803405572755 -->
<!-- iteration 21:  new_ratio = 1,6180339985218, old_ratio = 1,61803396316671, prev_ratio = 1,61803405572755 -->
<!-- f_next = 6765, base_values (0) = 10946 -->
<!-- f_next = 17711, base_values (1) = 17711 -->
<!-- new_ratio = 1,61803398501736,old_ratio = 1,6180339985218, prev_ratio = 1,61803396316671 -->
<!-- iteration 22:  new_ratio = 1,61803398501736, old_ratio = 1,6180339985218, prev_ratio = 1,61803396316671 -->
<!-- f_next = 10946, base_values (0) = 17711 -->
<!-- f_next = 28657, base_values (1) = 28657 -->
<!-- new_ratio = 1,6180339901756,old_ratio = 1,61803398501736, prev_ratio = 1,6180339985218 -->
<!-- iteration 23:  new_ratio = 1,6180339901756, old_ratio = 1,61803398501736, prev_ratio = 1,6180339985218 -->
<!-- f_next = 17711, base_values (0) = 28657 -->
<!-- f_next = 46368, base_values (1) = 46368 -->
<!-- new_ratio = 1,61803398820532,old_ratio = 1,6180339901756, prev_ratio = 1,61803398501736 -->
<!-- iteration 24:  new_ratio = 1,61803398820532, old_ratio = 1,6180339901756, prev_ratio = 1,61803398501736 -->
<!-- f_next = 28657, base_values (0) = 46368 -->
<!-- f_next = 75025, base_values (1) = 75025 -->
<!-- new_ratio = 1,6180339889579,old_ratio = 1,61803398820532, prev_ratio = 1,6180339901756 -->
<!-- iteration 25:  new_ratio = 1,6180339889579, old_ratio = 1,61803398820532, prev_ratio = 1,6180339901756 -->
<!-- f_next = 46368, base_values (0) = 75025 -->
<!-- f_next = 121393, base_values (1) = 121393 -->
<!-- new_ratio = 1,61803398867044,old_ratio = 1,6180339889579, prev_ratio = 1,61803398820532 -->
<!-- iteration 26:  new_ratio = 1,61803398867044, old_ratio = 1,6180339889579, prev_ratio = 1,61803398820532 -->
<!-- f_next = 75025, base_values (0) = 121393 -->
<!-- f_next = 196418, base_values (1) = 196418 -->
<!-- new_ratio = 1,61803398878024,old_ratio = 1,61803398867044, prev_ratio = 1,6180339889579 -->
<!-- iteration 27:  new_ratio = 1,61803398878024, old_ratio = 1,61803398867044, prev_ratio = 1,6180339889579 -->
<!-- f_next = 121393, base_values (0) = 196418 -->
<!-- f_next = 317811, base_values (1) = 317811 -->
<!-- new_ratio = 1,6180339887383,old_ratio = 1,61803398878024, prev_ratio = 1,61803398867044 -->
<!-- iteration 28:  new_ratio = 1,6180339887383, old_ratio = 1,61803398878024, prev_ratio = 1,61803398867044 -->
<!-- f_next = 196418, base_values (0) = 317811 -->
<!-- f_next = 514229, base_values (1) = 514229 -->
<!-- new_ratio = 1,61803398875432,old_ratio = 1,6180339887383, prev_ratio = 1,61803398878024 -->
<!-- iteration 29:  new_ratio = 1,61803398875432, old_ratio = 1,6180339887383, prev_ratio = 1,61803398878024 -->
<!-- f_next = 317811, base_values (0) = 514229 -->
<!-- f_next = 832040, base_values (1) = 832040 -->
<!-- new_ratio = 1,6180339887482,old_ratio = 1,61803398875432, prev_ratio = 1,6180339887383 -->
<!-- iteration 30:  new_ratio = 1,6180339887482, old_ratio = 1,61803398875432, prev_ratio = 1,6180339887383 -->
<!-- f_next = 514229, base_values (0) = 832040 -->
<!-- f_next = 1346269, base_values (1) = 1346269 -->
<!-- new_ratio = 1,61803398875054,old_ratio = 1,6180339887482, prev_ratio = 1,61803398875432 -->
<!-- iteration 31:  new_ratio = 1,61803398875054, old_ratio = 1,6180339887482, prev_ratio = 1,61803398875432 -->
<!-- f_next = 832040, base_values (0) = 1346269 -->
<!-- f_next = 2178309, base_values (1) = 2178309 -->
<!-- new_ratio = 1,61803398874965,old_ratio = 1,61803398875054, prev_ratio = 1,6180339887482 -->
<!-- iteration 32:  new_ratio = 1,61803398874965, old_ratio = 1,61803398875054, prev_ratio = 1,6180339887482 -->
<!-- f_next = 1346269, base_values (0) = 2178309 -->
<!-- f_next = 3524578, base_values (1) = 3524578 -->
<!-- new_ratio = 1,61803398874999,old_ratio = 1,61803398874965, prev_ratio = 1,61803398875054 -->
<!-- iteration 33:  new_ratio = 1,61803398874999, old_ratio = 1,61803398874965, prev_ratio = 1,61803398875054 -->
<!-- f_next = 2178309, base_values (0) = 3524578 -->
<!-- f_next = 5702887, base_values (1) = 5702887 -->
<!-- new_ratio = 1,61803398874986,old_ratio = 1,61803398874999, prev_ratio = 1,61803398874965 -->
<!-- iteration 34:  new_ratio = 1,61803398874986, old_ratio = 1,61803398874999, prev_ratio = 1,61803398874965 -->
<!-- f_next = 3524578, base_values (0) = 5702887 -->
<!-- f_next = 9227465, base_values (1) = 9227465 -->
<!-- new_ratio = 1,61803398874991,old_ratio = 1,61803398874986, prev_ratio = 1,61803398874999 -->
<!-- iteration 35:  new_ratio = 1,61803398874991, old_ratio = 1,61803398874986, prev_ratio = 1,61803398874999 -->
<!-- f_next = 5702887, base_values (0) = 9227465 -->
<!-- f_next = 14930352, base_values (1) = 14930352 -->
<!-- new_ratio = 1,61803398874989,old_ratio = 1,61803398874991, prev_ratio = 1,61803398874986 -->
<!-- iteration 36:  new_ratio = 1,61803398874989, old_ratio = 1,61803398874991, prev_ratio = 1,61803398874986 -->
<!-- f_next = 9227465, base_values (0) = 14930352 -->
<!-- f_next = 24157817, base_values (1) = 24157817 -->
<!-- new_ratio = 1,6180339887499,old_ratio = 1,61803398874989, prev_ratio = 1,61803398874991 -->
<!-- iteration 37:  new_ratio = 1,6180339887499, old_ratio = 1,61803398874989, prev_ratio = 1,61803398874991 -->
<!-- f_next = 14930352, base_values (0) = 24157817 -->
<!-- f_next = 39088169, base_values (1) = 39088169 -->
<!-- new_ratio = 1,61803398874989,old_ratio = 1,6180339887499, prev_ratio = 1,61803398874989 -->


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<TITLE>Fibonacci sequence</TITLE>

<script language="JavaScript">
<!-- begin

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	var my_mult = document.getElementById("multiplier");
	var my_base = document.getElementById("base");
	var my_iterations = document.getElementById("iterations");
	
	my_mult.value = p1;
	my_base.value = p2;
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<H2>Fibonacci sequence</H2>



<FORM method="POST" action="fibonacci.asp" enctype="application/x-www-form-urlencoded">

	Multiplier: <input type="text" id="multiplier" name="multiplier" title="For Fibonacci's sequence, set the multiplier to 1" value="1" size=2 onmouseover="showDiv ('multiplier_info')" onmouseout="hideDiv ('multiplier_info')">
	Base: <input type="text" id="base" name="base" title="For Fibonacci's sequence, set the base to 1 or 0,1" value="0,1" size=12 onmouseover="showDiv ('base_info')" onmouseout="hideDiv ('base_info')">
	Iterations: <input type="text" id="iterations" name="iterations" title="For Fibonacci's sequence, set the iterations to 41" value="37" size=3 onmouseover="showDiv ('iterations_info')" onmouseout="hideDiv ('iterations_info')">
<P>	<input type="button" value="Reset to Fibonacci" onclick="reset_parameters (1, 1, 41)">
	<input type="submit" value="Go">
	
</FORM>

<P>Next number in sequence = current number in sequence * multiplier + previous number in sequence

<H3>Result</H3>


<P>F(multiplier) = multiplier / 2 + Square root of ((multiplier / 2) * (multiplier / 2) + 1)
<P>F(1) = 1 / 2 + Square root of ((1/2) * (1/2) + 1)
<P>F(1) = 0,5 + Square root of (1,25)
<P>F(1) = 0,5 + SQR (1,25)

<H4>Prev=24157817, Next=39088169, 

ratio=1,61803398874989, 

inverse=0,618033988749895
</H4>


</DIV>






<DIV id="multiplier_info" style="position:absolute; top:10px; left:800px; visibility:hidden;">
<H3>Multiplier info</H3>
<P>The Multiplier is used only for the last number in the sequence when calculating the next number in the sequence.
<P>For example:
<UL>	<LI>with a base of (0,1) and multiplier = 3
	<LI>next number in sequence is: 1 * 3 + 0 = 3
	<LI>next number in sequence is: 3 * 3 + 1 = 10
	<LI>next number in sequence is: 10 * 3 + 3 = 33
	<LI>next number in sequence is: 33 * 3 + 10 = 109
	<LI>etc.
</UL>

<P>If the Base is greater than 2 numbers, then the multiplier is applied only to the last number in the sequence.
<P>For example:
<UL>	<LI>if the base is 4 numbers (0,1,2,3) and if multiplier = 2 then
	<LI>next number in sequence is : 3 * 2 + 2 + 1 + 0 = 9
	<LI>next number in sequence is: 9 * 2 + 3 + 2 + 1 = 24
	<LI>next number in sequence is: 24 * 2 + 9 + 3 + 2 = 62
	<LI>next number in sequence is: 62 * 2 + 24 + 9 + 3 = 160
	<LI>next number in sequence is: 160 * 2 + 62 + 24 + 9 = 415
	<LI>etc.
</UL>

</DIV>



<DIV id="base_info" style="position:absolute; top:10px; left:800px; visibility:hidden;">
<H3>Base info</H3>
<P>The Base is used to start the sequence.
<P>For a pure Fibonacci sequence, the base is 0,1.
<P>For variations, the base can be more that 2 numbers.
<P>For example:
<UL>	<LI>if the base is 4 numbers (0,1,2,3) [and multiplier is 1 - default value] then
	<LI>next number in sequence is : 3 + 2 + 1 + 0 = 6
	<LI>next number in sequence is: 6 + 3 + 2 + 1 = 12
	<LI>next number in sequence is: 12 + 6 + 3 + 2 = 23
	<LI>next number in sequence is: 23 + 12 + 6 + 3 = 44
	<LI>next number in sequence is: 44 + 23 + 12 + 6 = 85
	<LI>etc.
</UL>

<P>Note: the number of values is limited to 9.
<P>Note: if you specify a base of any 2 numbers, it will always result in the Fibonacci sequence. Try it !

</DIV>



<DIV id="iterations_info" style="position:absolute; top:10px; left:800px; visibility:hidden;">
<H3>Iterations info</H3>
<P>The Iterations is used to limit the number of times the sequence is calculated.
<P>For example:
<UL>	<LI>with a base of (0,1) and multiplier = 1 [default] and iterations = 4
	<LI>first number in sequence is: 1 * 1 + 0 = 1
	<LI>second number in sequence is: 1 * 1 + 1 = 2
	<LI>third number in sequence is: 2 * 1 + 1 = 3
	<LI>fourth number in sequence is: 3 * 1 + 2 = 5
</UL>

<P>Note: the number of iterations will stop when the ratio remains the same from one sequence to the next.
<P>Tip: You can view all the numbers calculated during the sequencing by viewing the page source code.

</DIV>


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