order 1 order 2 order 3 order 4 order 5 order 6 The Polish mathematician Wacław Sierpiński described the pattern in 1915, but it has appeared in Italian art since the 13th century. Though the Sierpinski triangle looks complex, it can be generated with a short recursive function. Your main task is to write a recursive function sierpinski() that plots a Sierpinski triangle of order n to standard drawing. Think recursively: sierpinski() Should draw one filled equilateral triangle (pointed downwards) and then call itself recursively three times (with an appropriate stopping condition). It should draw 1 filled triangle for n = 1; 4 filled triangles for n = 2; and 13 filled triangles for n = 3; and so forth.
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- The Polish mathematician Wacław Sierpiński described the pattern in 1915, but it has appeared in Italian art since the 13th century. Though the Sierpinski triangle looks complex, it can be generated with a short recursive function. Your main task is to write a recursive function sierpinski() that plots a Sierpinski triangle of order n to standard drawing. Think recursively: sierpinski() should draw one filled equilateral triangle (pointed downwards) and then call itself recursively three times (with an appropriate stopping condition). It should draw 1 filled triangle for n = 1; 4 filled triangles for n = 2; and 13 filled triangles for n = 3; and so forth. API specification. When writing your program, exercise modular design by organizing it into four functions, as specified in the following API: public class Sierpinski { // Height of an equilateral triangle whose sides are of the specified length. public static double height(double length) // Draws a filled equilateral…Pascal's triangle is a useful recursive definition that tells us the coefficients in the expansion of the polynomial (x + a)^n. Each element in the triangle has a coordinate, given by the row it is on and its position in the row (which you could call a column). Every number in Pascals triangle is defined as the sum of the item above it and the item above it and to the left. If there is a position that does not have an entry, we treat it as if we had a 0 there. *picture of the pascals triangle* Given the following recursive function signature, write the recursive function that takes a row and a column and finds the value at that position in the triangle. Assume that the triangle starts at row 0 and column 0. Examples: pascal(2, 1) -> 2, pascal(1, 2) -> 0 public int pascal(int row, int column) { }Write a RECURSIVE function, without using any loops, that prints the contents of a matrix with 3 columns. The function should take the matrix and the number of rows as arguments. void print_matrix(int arr[][3], int num_rows);
- Recursion can be direct or indirect. It is direct when a function calls itself and it is indirect recursion when a function calls another function that then calls the first function. To illustrate solving a problem using recursion, consider the Fibonacci series: - 1,1,2,3,5,8,13,21,34...The way to solve this problem is to examine the series carefully. The first two numbers are 1. Each subsequent number is the sum of the previous two numbers. Thus, the seventh number is the sum of the sixth and fifth numbers. More generally, the nth number is the sum of n - 2 and n - 1, as long as n > 2.Recursive functions need a stop condition. Something must happen to cause the program to stop recursing, or it will never end. In the Fibonacci series, n < 3 is a stop condition. The algorithm to use is this: 1. Ask the user for a position in the series.2. Call the fib () function with that position, passing in the value the user entered.3. The fib () function examines the argument (n). If n < 3…Write a recursive function that, given a sequence of comparable values, returns the count of elements where the current element is less than the following ( next ) element in the given sequence. See the examples given below. def count_ordered ( seq ) : """ Input : A sequence of comparable elements Output : The number of elements that are less than the following element in the sequence Example : >>> count_ordered ( [ 1 , 2 , 3 , 4 , 5 , 6 ] ) 5 >>> count_ordered ( ( 1 , 12, 7.3 , -2,4 ) ) 2 >>> count_ordered ( 'Python' ) 2 >>> count_ordered ( [ 6 ] ) 0 >>> count_ordered ( [ ] ) 0 """ In the first example above , count_ordered ( [ 1,2,3,4,5,6 ] )the returned answer is 5 because for all the first 5 numbers the current number is less than the next number. In the second example above, count_ordered ( ( 1,12,7.3 , -2,4 ) )the…The 4th problem mimics the situation where eagles flying in the sky can be spotted and counted.FindEagles: a recursive function that examines and counts the number of objects (eagles) in aphotograph. The data is in a two-dimensional grid of cells, each of which may be empty (value 0) orfilled (value 1 to 9). Maximum grid size is 50 x 50. The filled cells that are connected form an object(eagle). Two cells are connected if they are vertically, horizontally, or diagonally adjacent. Thefollowing figure shows 3 x 4 grids with 3 eagles. 0 0 1 21 0 0 01 0 3 1 FindEagle function takes as parameters the 2-D array and the x-y coordinates of a cell that is a part ofan eagle (non-zero value) and erases (change to 0) the image of an eagle. The function FindEagleshould return an integer value that counts how many cells has been counted as part of an eagle and havebeen erased. The following sample data has two pictures, the first one is 3 x 4, and the second one is 5 x 5 grids. Notethat your program…
- Write a recursive function that finds the minimum value in an ArrayList. Your function signature should be public static int findMinimum(ArrayList<Integer>) One way to think of finding a minimum recursively is to think “the minimum number is either the last element in the ArrayList, or the minimum value in the rest of the ArrayList”. For example, if you have the ArrayList [1, 3, 2, 567, 23, 45, 9], the minimum value in this ArrayList is either 9 or the minimum value in [1, 3, 2, 567, 23, 45] ================================================ import java.util.*; public class RecursiveMin{public static void main(String[] args){Scanner input = new Scanner(System.in);ArrayList<Integer> numbers = new ArrayList<Integer>();while (true){System.out.println("Please enter numbers. Enter -1 to quit: ");int number = input.nextInt();if (number == -1){break;}else {numbers.add(number);}} int minimum = findMinimum(numbers);System.out.println("Minimum: " + minimum);}public static int…For function decToBinary, write the missing parts of the recursion case. This function should return a string that stores the binary equivalent for int variable num. Example: The binary equivalent of 13 may be found by repeatedly dividing 13 by 2. So, 13 in base 2 is represented by the string "1101". Examples: decToBinary(13) -> "1101" public String decToBinary (int num) { if (num < 2) return Integer.toString(num); else return <<Missing recursive call>> + <<Missing calculation>>;}The Tower of Hanoi is a puzzle where n disks of different sizes arestacked in ascending order on one rod and there are two other rods with nodisks on them. The objective is to move all disks from the first rod to thethird, such that:- only one disk is moved at a time- a larger disk can never be placed on top of a smaller oneWrite a recursive function that outputs the sequence of steps needed tosolve the puzzle with n disks.Write a test program in C++ that allows the user to input number of disks andthen uses your function to output the steps needed to solve the puzzle.Hint: If you could move up n−1 of the disks from the first post to thethird post using the second post as a spare, the last disk could be moved fromthe first post to the second post. Then by using the same technique you canmove the n−1 disks from the third post to the second post, using the firstdisk as a spare. There! You have the puzzle solved. You only have to decidewhat the nonrecursive case is, what the recursive…
- Java, I am not displaying my results correct. It should add up the digits in the string. input string |result for the sumIt Recursion functions and a findMax function that finds the largest number in a string "1d2d3d" | 6 total "55" |10 total "xx" | 0 total "12x8" |12 Max number "012x88" |88 Max Number "012x88ttttt9xe33ppp100" |100 Max Number public class Finder { //Write two recursive functions, both of which will parse any length string that consists of digits and numbers. Both functions //should be…This section is divided into two subsections. Each subsection contains a problem to be solved using both recursive and non-recursive approaches. You need to implement the solution in one of your preferred languages ( C++, or JAVA). In addition, you need to provide the order growth analysis and plot the input size vs. run-time graph for both approaches. The n th Triangle Problem Write a code for finding the n th triangle number of triangle sequences: 1, 3, 6, 10, ..., n. That is, your code should accept an integer number, which indicates the triangle levels, and returns how many dots we need to form a triangle with respect to the given level. For example, consider the Fig 1. For n = 3 (can be also written as T3), your code should returns 6. Provide a single program consists of the following: • Write a function called TriangularRecursive for the recursive version that takes number of level as an int argument. Hints: 1) Identify the base case for the TriangularRecursive function. 2) Let…Write a recursive function that find the minimum element in an array of integers.int findMin(int a[], int size);