Generating Sequences

JavaScript/Generating-sequences.md

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+# Generating-sequences
+
+Mathematical sequences occur so frequently in nature very often this transfers into computer programming problems. For example, the 
+cicadas who appear periodically but only emerge after a prime number of years.  Or sunflowers which have either 34, 55 or 89 petals 
+on their face, depending upon the size of the sunflower. Additionally, lilies and irises have three petals, buttercups and wild roses 
+have five, delphiniums have eight petals, all related to the Fibonacci sequence. Or simply triangular numbers in bowling! 
+We are surrounded by sequences which are regularly analysed within computer programming.
+
+Being equipped to generate such sequences is vital. This is dedicated to exploring the possibleways these sequences can be produced 
+within a computer program using javascript:
+
+Prime Numbers
+
+A prime number is a natural number that has exactly two distinct natural number divisors: 1 and itself. There are several prime number
+sieves in circulation, however, Sieve of Eratosthenes is a simple ancient algorithm.
+
+By iterating through an array of numbers starting at the beginning and removing all subsequent numbers that are a factor of the prime.
+The multiples of a given prime are generated as a sequence of numbers starting from that prime, with constant difference between
+them that is equal to that prime. This is the sieve's key distinction from using trial division to sequentially test each 
+candidate number for divisibility by each prime:
+
+	function primes(num) {
+		var x = [];
+		for( var i =0; i<= num; i++) {
+			x[i] = true;
+		}
+		x[0]= false;
+		x[1]= false;
+		x[2]= true;
+		for ( var j = 2; j< num; j++) {
+			for (var k = 2; j*k <= num; k++) {
+				x[j*k] = false;
+			}
+		}
+
+		primeValues = [];
+
+	  for (var i = 0; i< num; i ++) {
+	    if (x[i] == true) { 
+	      primeValues.push(i);
+	    }
+	  }
+	return primeValues;
+	}
+
+	primes(20);
+	[2, 3, 5, 7, 11, 13, 17, 19]
+
+Fibonnaci Sequence
+
+In mathematical terms, the sequence Fn of Fibonacci numbers is defined by the recurrence relation Fn = Fn-1 + Fn-2 with seed values.
+
+	function fibonacci(nums) {
+
+		var fibSequence = [];
+
+		fibSequence[0] = 0;
+		fibSequence[1] = 1;
+
+		for (var i = 2; i < nums; i ++) {
+			fibSequence[i] = fibSequence[i-1] + fibSequence[i-2];
+		}
+
+		return fibSequence;
+	}
+
+	fibonacci(8);
+	[ 0, 1, 1, 2, 3, 5, 8, 13 ]