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# Hosoya’s Triangle

The Fibonacci triangle or Hosoya’s triangle is a triangular arrangement of numbers based on Fibonacci numbers. Each number is the sum of two numbers above in either the left diagonal or the right diagonal. The first few rows are:

The numbers in this triangle follow the recurrence relations

Relation to Fibonacci numbers
The entries in the triangle satisfy the identity

Thus, the two outermost diagonals are the Fibonacci numbers, while the numbers on the middle vertical lines are the squares of the Fibonacci numbers. All the other numbers in the triangle are the product of two distinct Fibonacci numbers greater than 1. The row sums are the first convolved Fibonacci numbers.
Sources: Stackoverflow, Wikipedia
Given a positive integers n. The task is print Hosoya’s triangle of size n.
Examples:

```Input : n = 4
Output :
1
1 1
2 1 2
3 2 2 3

Input : n = 5
Output :
1
1 1
2 1 2
3 2 2 3
5 3 4 3 5```

Below is the implementation of printing Hosoya’s triangle of height n:

## C++

 `// CPP Program to print Hosoya's` `// triangle of height n.` `#include ` `using` `namespace` `std;`   `int` `Hosoya(``int` `n, ``int` `m)` `{` `    ``// Base case` `    ``if` `((n == 0 && m == 0) ||` `        ``(n == 1 && m == 0) || ` `        ``(n == 1 && m == 1) ||` `        ``(n == 2 && m == 1))` `        ``return` `1;`   `    ``// Recursive step` `    ``if` `(n > m)` `        ``return` `Hosoya(n - 1, m) ` `               ``+ Hosoya(n - 2, m);`   `    ``else` `if` `(m == n)` `        ``return` `Hosoya(n - 1, m - 1) ` `               ``+ Hosoya(n - 2, m - 2);`   `    ``else` `        ``return` `0;` `}`   `// Print the Hosoya triangle of height n.` `void` `printHosoya(``int` `n)` `{` `    ``for` `(``int` `i = 0; i < n; i++) {` `        ``for` `(``int` `j = 0; j <= i; j++) ` `            ``cout << Hosoya(i, j) << ``" "``;     `   `        ``cout << endl;` `    ``}` `}`   `// Driven Program` `int` `main()` `{` `    ``int` `n = 5;` `    ``printHosoya(n);` `    ``return` `0;` `}`

## Java

 `// Java Program to print Hosoya's ` `// triangle of height n.` `import` `java.util.*;`   `class` `GFG {` `    `  `    ``static` `int` `Hosoya(``int` `n, ``int` `m)` `    ``{` `        ``// Base case` `        ``if` `((n == ``0` `&& m == ``0``) ||` `            ``(n == ``1` `&& m == ``0``) || ` `            ``(n == ``1` `&& m == ``1``) || ` `            ``(n == ``2` `&& m == ``1``))` `            ``return` `1``;` `     `  `        ``// Recursive step` `        ``if` `(n > m)` `            ``return` `Hosoya(n - ``1``, m)` `                   ``+ Hosoya(n - ``2``, m);` `            `  `        ``else` `if` `(m == n)` `            ``return` `Hosoya(n - ``1``, m - ``1``)` `                    ``+ Hosoya(n - ``2``, m - ``2``);` `            `  `        ``else` `            ``return` `0``;` `    ``}` `     `  `    ``// Print the Hosoya triangle of height n.` `    ``static` `void` `printHosoya(``int` `n)` `    ``{` `        ``for` `(``int` `i = ``0``; i < n; i++)` `        ``{` `            ``for` `(``int` `j = ``0``; j <= i; j++) ` `                ``System.out.print(Hosoya(i, j) ` `                                        ``+ ``" "``);       ` `     `  `            ``System.out.println(``""``);` `        ``}` `    ``}`   `    `  `    ``/* Driver program to test above function */` `    ``public` `static` `void` `main(String[] args) ` `    ``{` `        ``int` `n = ``5``;` `        ``printHosoya(n);` `         `  `    ``}` `}`   `// This code is contributed by  Arnav Kr. Mandal.`

## Python3

 `# Python3 code to print Hosoya's` `# triangle of height n.`   `def` `Hosoya( n , m ):`   `    ``# Base case` `    ``if` `((n ``=``=` `0` `and` `m ``=``=` `0``) ``or` `        ``(n ``=``=` `1` `and` `m ``=``=` `0``) ``or` `        ``(n ``=``=` `1` `and` `m ``=``=` `1``) ``or` `        ``(n ``=``=` `2` `and` `m ``=``=` `1``)):` `                ``return` `1` `    `  `    ``# Recursive step` `    ``if` `n > m:` `        ``return` `Hosoya(n ``-` `1``, m) ` `                    ``+` `Hosoya(n ``-` `2``, m)`   `    ``elif` `m ``=``=` `n:` `        ``return` `Hosoya(n ``-` `1``, m ``-` `1``) ` `                        ``+` `Hosoya(n ``-` `2``,    m ``-` `2``)`   `    ``else``:` `        ``return` `0` `        `  `# Print the Hosoya triangle of height n.` `def` `printHosoya( n ):` `    ``for` `i ``in` `range``(n):` `        ``for` `j ``in` `range``(i ``+` `1``):` `            ``print``(Hosoya(i, j) , end ``=` `" "``)` `        ``print``(``"\n"``, end ``=` `"")` `        `  `# Driven Code` `n ``=` `5` `printHosoya(n)`   `# This code is contributed by Sharad_Bhardwaj`

## C#

 `// C# Program to print Hosoya's ` `// triangle of height n.` `using` `System;`   `class` `GFG {` `    `  `    ``static` `int` `Hosoya(``int` `n, ``int` `m)` `    ``{` `        ``// Base case` `        ``if` `((n == 0 && m == 0) ||` `            ``(n == 1 && m == 0) || ` `            ``(n == 1 && m == 1) || ` `            ``(n == 2 && m == 1))` `            ``return` `1;` `    `  `        ``// Recursive step` `        ``if` `(n > m)` `            ``return` `Hosoya(n - 1, m)` `                 ``+ Hosoya(n - 2, m);` `            `  `        ``else` `if` `(m == n)` `            ``return` `Hosoya(n - 1, m - 1)` `                 ``+ Hosoya(n - 2, m - 2);` `            `  `        ``else` `            ``return` `0;` `    ``}` `    `  `    ``// Print the Hosoya triangle of height n.` `    ``static` `void` `printHosoya(``int` `n)` `    ``{` `        ``for` `(``int` `i = 0; i < n; i++)` `        ``{` `            ``for` `(``int` `j = 0; j <= i; j++) ` `                ``Console.Write(Hosoya(i, j) ` `                                        ``+ ``" "``); ` `    `  `            ``Console.WriteLine(``""``);` `        ``}` `    ``}`   `    `  `    ``/* Driver program to test above function */` `    ``public` `static` `void` `Main() ` `    ``{` `        ``int` `n = 5;` `        `  `        ``printHosoya(n);` `        `  `    ``}` `}`   `// This code is contributed by vt_m.`

## PHP

 ` ``\$m``)` `        ``return` `Hosoya(``\$n` `- 1,``\$m``) + ` `               ``Hosoya(``\$n` `- 2, ``\$m``);`   `    ``else` `if` `(``\$m` `== ``\$n``)` `        ``return` `Hosoya(``\$n` `- 1, ``\$m` `- 1) + ` `               ``Hosoya(``\$n` `- 2, ``\$m` `- 2);`   `    ``else` `        ``return` `0;` `}`   `// Print the Hosoya ` `// triangle of height n.` `function` `printHosoya( ``\$n``)` `{` `    ``for` `( ``\$i` `= 0; ``\$i` `< ``\$n``; ``\$i``++)` `    ``{` `        ``for` `( ``\$j` `= 0; ``\$j` `<= ``\$i``; ``\$j``++) ` `            ``echo` `Hosoya(``\$i``, ``\$j``) , ``" "``; ` `            ``echo` `"\n"``;` `    ``}` `}`   `// Driven Code` `\$n` `= 5;` `printHosoya(``\$n``);`   `// This code is contributed by anuj_67.` `?>`

## Javascript

 ``

Output:

```1
1 1
2 1 2
3 2 2 3
5 3 4 3 5 ```

Time Complexity: O(n2)

Auxiliary Space: O(n2)
Below is the implementation of printing Hosoya’s triangle of height n using Dynamic Programming:

## C++

 `// CPP Program to print Hosoya's triangle of height n.` `#include ` `#define N 5` `using` `namespace` `std;`   `// Print the Hosoya triangle of height n.` `void` `printHosoya(``int` `n)` `{` `    ``int` `dp[N][N];` `    ``memset``(dp, 0, ``sizeof``(dp));`   `    ``// base case.` `    ``dp[0][0] = dp[1][0] = dp[1][1] = 1;`   `    ``// For each row.` `    ``for` `(``int` `i = 2; i < n; i++) {`   `        ``// for each column;` `        ``for` `(``int` `j = 0; j < n; j++) {`   `            ``// recursive steps.` `            ``if` `(i > j)` `                ``dp[i][j] = dp[i - 1][j] + dp[i - 2][j];`   `            ``else` `                ``dp[i][j] = dp[i - 1][j - 1] + dp[i - 2][j - 2];` `        ``}` `    ``}`   `    ``// printing the solution` `    ``for` `(``int` `i = 0; i < n; i++) {` `        ``for` `(``int` `j = 0; j <= i; j++) ` `            ``cout << dp[i][j] << ``" "``;        `   `        ``cout << endl;` `    ``}` `}`   `// Driven Program` `int` `main()` `{` `    ``int` `n = 5;` `    ``printHosoya(n);` `    ``return` `0;` `}`

## Java

 `// JAVA Code for Hosoya Triangle` `import` `java.util.*;`   `class` `GFG {` `    `  `    ``static` `int` `N = ``5``;` `    `  `    ``// Print the Hosoya triangle ` `    ``// of height n.` `    ``static` `void` `printHosoya(``int` `n)` `    ``{` `        ``int` `dp[][] = ``new` `int``[N][N];` `        `  `        ``// base case.` `        ``dp[``0``][``0``] = dp[``1``][``0``] = ``1``;` `        ``dp[``1``][``1``] = ``1``;` `     `  `        ``// For each row.` `        ``for` `(``int` `i = ``2``; i < n; i++) ` `        ``{` `            ``// for each column;` `            ``for` `(``int` `j = ``0``; j < n; j++) ` `            ``{` `                 ``// recursive steps.` `                ``if` `(i > j)` `                    ``dp[i][j] = dp[i - ``1``][j] + ` `                                        ``dp[i - ``2``][j];` `     `  `                ``else` `                    ``dp[i][j] = dp[i - ``1``][j - ``1``] +` `                                    ``dp[i - ``2``][j - ``2``];` `            ``}` `        ``}` `     `  `        ``// printing the solution` `        ``for` `(``int` `i = ``0``; i < n; i++) ` `        ``{` `            ``for` `(``int` `j = ``0``; j <= i; j++) ` `                ``System.out.print(dp[i][j] + ``" "``);        ` `     `  `            ``System.out.println(``""``);` `        ``}` `    ``}` `    `  `    ``/* Driver program*/` `    ``public` `static` `void` `main(String[] args) ` `    ``{` `        ``int` `n = ``5``;` `        ``printHosoya(n);` `    ``}` `}`   `// This code is contributed by Arnav Kr. Mandal.`

## Python3

 `# Python3 Program to print ` `# Hosoya's triangle of height n.` `N ``=` `5`   `# Print the Hosoya triangle ` `# of height n.` `def` `printHosoya(n):` `    ``dp ``=` `[[``0` `for` `i ``in` `range``(N)] ` `             ``for` `i ``in` `range``(N)]` `             `  `    ``# base case.` `    ``dp[``0``][``0``] ``=` `dp[``1``][``0``] ``=` `dp[``1``][``1``] ``=` `1` `    `  `    ``# For each row.` `    ``for` `i ``in` `range``(``2``, n):` `        `  `        ``# for each column` `        ``for` `j ``in` `range``(n):` `            `  `            ``# recursive steps.` `            ``if` `(i > j):` `                ``dp[i][j] ``=` `(dp[i ``-` `1``][j] ``+` `                            ``dp[i ``-` `2``][j])` `            ``else``:` `                ``dp[i][j] ``=` `(dp[i ``-` `1``][j ``-` `1``] ``+` `                            ``dp[i ``-` `2``][j ``-` `2``])` `                            `  `    ``# printing the solution ` `    ``for` `i ``in` `range``(n):` `        ``for` `j ``in` `range``(i ``+` `1``):` `            ``print``(dp[i][j], end ``=` `' '``)` `        ``print``()`   `# Driver Code` `n ``=` `5` `printHosoya(n)`   `# This code is contributed ` `# by sahilshelangia`

## C#

 `// C# Code for Hosoya Triangle` `using` `System;`   `class` `GFG {` `    `  `    ``static` `int` `N = 5;` `    `  `    ``// Print the Hosoya triangle ` `    ``// of height n.` `    ``static` `void` `printHosoya(``int` `n)` `    ``{` `        ``int` `[,]dp = ``new` `int``[N,N];` `        `  `        ``// base case.` `        ``dp[0,0] = dp[1,0] = 1;` `        ``dp[1,1] = 1;` `    `  `        ``// For each row.` `        ``for` `(``int` `i = 2; i < n; i++) ` `        ``{` `            ``// for each column;` `            ``for` `(``int` `j = 0; j < n; j++) ` `            ``{` `                ``// recursive steps.` `                ``if` `(i > j)` `                    ``dp[i,j] = dp[i - 1,j] + ` `                              ``dp[i - 2,j];` `    `  `                ``else` `                    ``dp[i,j] = dp[i - 1,j - 1] ` `                           ``+ dp[i - 2,j - 2];` `            ``}` `        ``}` `    `  `        ``// printing the solution` `        ``for` `(``int` `i = 0; i < n; i++) ` `        ``{` `            ``for` `(``int` `j = 0; j <= i; j++) ` `                ``Console.Write(dp[i,j] + ``" "``); ` `    `  `            ``Console.WriteLine(``""``);` `        ``}` `    ``}` `    `  `    ``/* Driver program*/` `    ``public` `static` `void` `Main() ` `    ``{` `        ``int` `n = 5;` `        `  `        ``printHosoya(n);` `    ``}` `}`   `// This code is contributed by Vt_m.`

## PHP

 ` ``\$j``)` `                ``\$dp``[``\$i``][``\$j``] = ``\$dp``[``\$i` `- 1][``\$j``] ` `                            ``+ ``\$dp``[``\$i` `- 2][``\$j``];`   `            ``else` `                ``\$dp``[``\$i``][``\$j``] = ``\$dp``[``\$i` `- 1][``\$j` `- 1] ` `                            ``+ ``\$dp``[``\$i` `- 2][``\$j` `- 2];` `        ``}` `    ``}`   `    ``// printing the solution` `    ``for` `(``\$i` `= 0; ``\$i` `< ``\$n``; ``\$i``++) {` `        ``for` `(``\$j` `= 0; ``\$j` `<= ``\$i``; ``\$j``++) ` `            ``echo` `\$dp``[``\$i``][``\$j``].``" "``; `   `        ``echo` `"\n"``;` `    ``}` `}`   `// Driven Program`   `    ``\$n` `= 5;` `    ``printHosoya(``\$n``);`   `// This code is contributed by mits` `?>`

## Javascript

 `    `

Output:

```1
1 1
2 1 2
3 2 2 3
5 3 4 3 5 ```

Time complexity: O(n*n)

space complexity: O(n*n)

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