Merge branch 'main' of https://github.com/Twiggecode/Integer-Sequences into main

Author: beamandala

Diff for: Alcuin Sequence/AlcuinSequence.c

@@ -0,0 +1,30 @@
+#include<stdio.h>
+#include<math.h>
+
+/**
+ * Alcuin Sequence
+ * Formula: a(n) = round(n^2/12) – floor(n/4)*floor((n+2)/4)
+ * Example: 0, 0, 0, 1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 7 ...
+ */
+
+// Algorithm for calculating Alcuin numbers
+int Alcuin(int n)
+{
+    double _n = n, val;
+ 
+    val = round((_n * _n) / 12)
+          - floor(_n / 4)
+                * floor((_n + 2) / 4);
+ 
+    return val;
+}
+ 
+// Main driver
+int main()
+{
+    int n;
+    printf("Enter the nth valu to find: ");
+    scanf("%d",&n);
+    printf("The term at the required index of the Alcuin Sequence is %d",Alcuin(n-1));
+    return 0;
+}

Diff for: Alcuin Sequence/AlcuinSequence.cpp

@@ -0,0 +1,30 @@
+#include <bits/stdc++.h>
+using namespace std;
+/**
+ * Alcuin Sequence
+ * Formula: a(n) = round(n^2/12) – floor(n/4)*floor((n+2)/4)
+ * Example: 0, 0, 0, 1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 7 ...
+ */
+
+// Algorithm for calculating Alcuin numbers
+int Alcuin(int n)
+{
+    double _n = n, val;
+ 
+    val = round((_n * _n) / 12)
+          - floor(_n / 4)
+                * floor((_n + 2) / 4);
+ 
+    return val;
+}
+ 
+// Main driver
+int main()
+{
+    int n;
+    cout<< "Enter the nth value to find: ";
+    cin>> n;
+    cout<< "The term at the required index of the Alcuin Sequence is ";
+    cout<<Alcuin(n-1);
+    return 0;
+}

Diff for: Alcuin Sequence/AlcuinSequence.py

@@ -5,6 +5,9 @@
 
 
 def alcuin(n):
+    n = (
+        n - 1
+    )  # Edit: The sequence begins with the coefficient of x^0, first term requires n = 0
     return round((n * n) / 12) - floor(n / 4) * floor((n + 2) / 4)
 
 

Diff for: Alcuin Sequence/Alcuin_Sequence.java

@@ -15,7 +15,7 @@ public static void main(String[] args) {
     }
 
     private static int Alcuin(int _n) {
-        double n = _n;
+        double n = _n-1; //Edit: The sequence begins with the coefficient of x^0, first term requires n = 0
         return (int)(Math.round((n * n) / 12) - Math.floor(n / 4) * Math.floor((n + 2) / 4));
     }
 }

Diff for: All Exponent Integer Series/pow(x,y).c

@@ -0,0 +1,37 @@
+// C program for calculating x raised to the power of y 
+// Time complexity O(log(n))
+
+#include <stdio.h>
+
+typedef unsigned long long U64;
+
+U64 power(U64 x, U64 y)
+{
+    U64 result = 1;
+    while (y > 0) {
+        if (y % 2 == 0) // y is even
+        {
+            x = x * x;
+            y = y / 2;
+        }
+        else // y isn't even
+        {
+            result = result * x;
+            y = y - 1;
+        }
+    }
+    return result;
+}
+ 
+// Driver Code
+int main()
+{   
+    U64 x,y;
+    printf("Enter x in x^y: ");
+    scanf("%llu", &x);
+    printf("Enter y in x^y: ");
+    scanf("%llu",&y);
+    printf("%llu to power %llu is: %llu ",x,y,power(x,y));
+    return 0;
+}
+ 

Diff for: All Exponent Integer Series/pow(x,y).cpp

@@ -3,10 +3,12 @@
 // Time complexity O(log(n))
 #include <bits/stdc++.h>
 using namespace std;
- 
-int power(int x, int y)
+
+typedef unsigned long long U64;
+
+U64 power(U64 x, U64 y)
 {
-    int result = 1;
+    U64 result = 1;
     while (y > 0) {
         if (y % 2 == 0) // y is even
         {
@@ -25,11 +27,11 @@ int power(int x, int y)
 // Driver Code
 int main()
 {   
-    # just set the x,y to suit your needs
-    int x,y;
-    cin>>x>>y;
-  
- 
+    U64 x,y;
+    cout<<"Enter x in x^y: ";
+    cin>>x;
+    cout<<"Enter y in x^y: ";
+    cin>>y;
     cout << (power(x, y));
     return 0;
 }

Diff for: All Exponent Integer Series/pow(x,y).py

@@ -0,0 +1,5 @@
+#/usr/bin/python3
+
+x=int(input("Enter x in x^y: "))
+y=int(input("Enter y in x^y: "))
+print(str(x)+"^"+str(y)+" is: "+str(pow(x,y)))

Diff for: Factorials/factorials.cpp

@@ -0,0 +1,50 @@
+#include<iostream>
+using namespace std;
+
+// Define the number of digits the output factorial can contain
+#define HUGE 10000 
+
+// High school multiplication - storing output as individual digits in an array
+// Multiplying each digit with individually and adjusting carry overs
+int mult(int x, int answer[], int output_digits)
+{
+    int carry = 0;  
+ 
+    for (int i=0; i<output_digits; i++)
+    {
+        int prod = answer[i] * x + carry;
+        answer[i] = prod % 10; 
+        carry  = prod/10;   
+    }
+ 
+    while (carry)
+    {
+        answer[output_digits] = carry%10;
+        carry = carry/10;
+        output_digits++;
+    }
+    return output_digits;
+}
+// Factorial function
+void factorial(int n)
+{
+    int answer[HUGE];
+    answer[0] = 1;
+    int output_digits = 1;
+
+    for (int x=2; x<=n; x++)
+        output_digits = mult(x, answer, output_digits);
+ 
+    cout << "Factorial of the given number is: \n";
+    for (int i=output_digits-1; i>=0; i--)
+        cout << answer[i];
+}
+// Main Driver
+int main()
+{
+    int n;
+    cout<<"Enter a number: ";
+    cin>>n;
+    factorial(n);
+    return 0;
+}

Diff for: Factorials/factorials.py

@@ -1,19 +1,43 @@
-def factorial(a: int):
+"""
+In python default recursion limit is 1000. 
+Naive recursive implementation will raise RecursionError for numbers >= 1000.
+By using memoization we can calculate factorials of those numbers 
+step by step as shown in the main block.
+"""
+
+
+def factorial(a: int, results={}):
     """
     handle edge case
     a == 0 or 1 or -ve number
     """
+    if a in results:
+        return results[a]
+
     if a == 0 or a == 1:
         return 1
 
     if a < 0:
         return "Negative Integer is not allowed"
 
     # recursive call
-    return a * factorial(a - 1)
+    result = a * factorial(a - 1)
+
+    # memoize the result
+    results[a] = result
+
+    return result
 
 
 if __name__ == "__main__":
-    n = int(input("Enter the positive integer value: "))
+    n = 9
+    answer = factorial(n)
+    print(f"{n}! = {answer}")
+
+    n = 900
+    answer = factorial(n)
+    print(f"{n}! = {answer}")
+
+    n = 999
     answer = factorial(n)
     print(f"{n}! = {answer}")

Diff for: Fibonacci Sequence/fibonacci.py

@@ -1,32 +1,69 @@
-# This code will return the nth element of the Fibonacci sequence
+"""
+Calculate nth Fibonacci term
+"""
 
 
-def fibonacci(n):
-    n += 1
-    i = 1
-    num1, num2 = 0, 1
+class NegativeNumber(Exception):
+    pass
 
-    if n == 1:
+
+def fibonacci_iter(n):
+    """Iterative implementation of Fibonacci"""
+    a, b = 0, 1
+
+    if n < 0:
+        raise NegativeNumber("Fibonacci not defined for negative numbers!")
+
+    if n == 0:
         return 0
-    elif n == 2:
-        return 1
-    else:
 
-        while i < n - 1:
+    for _ in range(1, n):
+        a, b = b, a + b
+
+    return b
+
+
+def fibonacci_naive_recur(n):
+    """Naive recursive computation of Fibonacci.
+    Time complexity is O(1.6180 ^ n). It's big!"""
+
+    if n < 0:
+        raise NegativeNumber("Fibonacci not defined for negative numbers!")
+
+    if n < 2:
+        return n
+
+    return fibonacci_naive_recur(n - 1) + fibonacci_naive_recur(n - 2)
+
+
+def fibonacci(n, cache={}):
+    """Memoized recursive computation of Fibonacci.
+    Time complexity is O(n). Better!"""
+    if n < 0:
+        raise NegativeNumber("Fibonacci not defined for negative numbers!")
+
+    if n < 2:
+        return n
+
+    if n in cache:
+        return cache[n]
+
+    cache[n] = fibonacci(n - 1) + fibonacci(n - 2)
 
-            sum = num1 + num2
-            num1, num2 = num2, sum
-            i += 1
-        return sum
+    return cache[n]
 
 
-while True:
-    try:
-        my_num = int(input("Enter a number:"))
-        if my_num > -1:
-            break
-    except:
-        print("Enter a positive integer: ")
+def tests(fib):
+    assert fib(0) == 0
+    assert fib(1) == 1
+    assert fib(2) == 1
+    assert fib(3) == 2
+    assert fib(4) == 3
+    assert fib(5) == 5
 
 
-print(f"The {my_num}th element of fibonacci series is:", fibonacci(my_num))
+if __name__ == "__main__":
+    tests(fibonacci_iter)
+    tests(fibonacci_naive_recur)
+    tests(fibonacci)
+    print(fibonacci_iter(10000))

Diff for: Gould Sequence/GouldSequence.cpp

@@ -0,0 +1,17 @@
+#include<iostream>
+using namespace std;
+
+int main()
+{
+    int n,inp,count=0;
+    cout<<"Enter the nth value: ";
+    cin>>n;
+    inp=n;
+    while(inp)
+    {
+        count+=inp&1;
+        inp>>=1;
+    }
+    cout<<n<<"th row in the pascal triangle has "<<(1<<count)<<" odd numbers!";
+    return 0;
+}