# Gould’s Sequence

Given a positive Integer n, The task is to print Gould’s sequence up to nth term.

In Mathematics, Gould’s sequence is an integer sequence whose nth term tells the count of odd numbers in n-1th row of the pascal triangle. This sequence is consist of only power’s of 2.

For Example:

```Row Number                Pascal's triangle                 count of odd numbers in ith row
0th row                             1                                         1
1st row                           1   1                                       2
2nd row                         1   2   1                                     2
3rd row                       1   3   3   1                                   4
4th row                     1   4   6   4   1                                 2
5th row                   1   5   10  10  5   1                               4
6th row                 1   6   15  20  15  6   1                             4
7th row               1   7   21  35  35  21  7   1                           8
8th row             1  8   28   56  70   56  28  8  1                         2
9th row           1   9  36  84  126  126  84  36  9  1                       4
10th row        1  10  45  120 210  256  210 120 45 10  1                     4
```

So first few terms of Gould’s sequence are-

1, 2, 2, 4, 2, 4, 4, 8, 2, 4, 4, 8, 4, 8, 8, 16, 2, 4, 4, 8, 4, 8, 8, 16, 4, 8, 8, 16, 8, 16, 16, 32

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

A Simple Solution to generate Gould’s sequence is to generate every row of pascal’s triangle from 0th row to nth row and count odd numbers appearing in each row.
ith element of nth row can be calculated easily by calculating binomial coefficient C(n, i).
Below is the implementation of above idea

## C++

 `// CPP program to generate ` `// Gould's Sequence ` ` `  `#include ` `using` `namespace` `std; ` ` `  `// Function to generate gould's Sequence ` `void` `gouldSequence(``int` `n) ` `{ ` `    ``// loop to generate each row ` `    ``// of pascal's Triangle up to nth row ` `    ``for` `(``int` `row_num = 1; row_num <= n; row_num++) { ` ` `  `        ``int` `count = 1; ` `        ``int` `c = 1; ` ` `  `        ``// Loop to generate each element of ith row ` `        ``for` `(``int` `i = 1; i <= row_num; i++) { ` ` `  `            ``c = c * (row_num - i) / i; ` ` `  `            ``// if c is odd ` `            ``// increment count ` `            ``if` `(c % 2 == 1) ` `                ``count++; ` `        ``} ` ` `  `        ``// print count of odd elements ` `        ``cout << count << ``" "``; ` `    ``} ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` ` `  `    ``// Get n ` `    ``int` `n = 16; ` ` `  `    ``// Function call ` `    ``gouldSequence(n); ` ` `  `    ``return` `0; ` `} `

## Java

 `// JAVA program to generate ` `// Gould's Sequence ` ` `  `class` `GFG { ` ` `  `    ``// Function to generate gould's Sequence ` `    ``static` `void` `gouldSequence(``int` `n) ` `    ``{ ` `        ``// loop to generate each row ` `        ``// of pascal's Triangle up to nth row ` `        ``for` `(``int` `row_num = ``1``; row_num <= n; row_num++) { ` ` `  `            ``int` `count = ``1``; ` `            ``int` `c = ``1``; ` ` `  `            ``// Loop to generate each element of ith row ` `            ``for` `(``int` `i = ``1``; i <= row_num; i++) { ` ` `  `                ``c = c * (row_num - i) / i; ` ` `  `                ``// if c is odd ` `                ``// increment count ` `                ``if` `(c % ``2` `== ``1``) ` `                    ``count++; ` `            ``} ` ` `  `            ``// print count of odd elements ` `            ``System.out.print(count + ``" "``); ` `        ``} ` `    ``} ` ` `  `    ``// Driver code ` `    ``public` `static` `void` `main(String[] args) ` `    ``{ ` ` `  `        ``// Get n ` `        ``int` `n = ``16``; ` ` `  `        ``// Function call ` `        ``gouldSequence(n); ` `    ``} ` `} `

## Python 3

 `# Python 3 program to generate ` `# Gould's Sequence ` ` `  `# Function to generate gould's Sequence ` `def` `gouldSequence(n): ` ` `  `    ``# loop to generate each row ` `    ``# of pascal's Triangle up to nth row ` `    ``for` `row_num ``in` `range` `(``1``, n):  ` `     `  `        ``count ``=` `1` `        ``c ``=` `1` ` `  `        ``# Loop to generate each  ` `        ``# element of ith row ` `        ``for` `i ``in` `range` `(``1``, row_num): ` `            ``c ``=` `c ``*` `(row_num ``-` `i) ``/` `i ` ` `  `            ``# if c is odd ` `            ``# increment count ` `            ``if` `(c ``%` `2` `=``=` `1``): ` `                ``count ``+``=` `1` ` `  `        ``# print count of odd elements ` `        ``print``(count, end ``=` `" "``) ` `         `  `# Driver code ` ` `  `# Get n ` `n ``=` `16``; ` ` `  `# Function call ` `gouldSequence(n) ` ` `  `# This code is contributed  ` `# by Akanksha Rai `

## C#

 `// C# program to generate ` `// Gould's Sequence ` ` `  `using` `System; ` `class` `GFG { ` ` `  `    ``// Function to generate gould's Sequence ` `    ``static` `void` `gouldSequence(``int` `n) ` `    ``{ ` `        ``// loop to generate each row ` `        ``// of pascal's Triangle up to nth row ` `        ``for` `(``int` `row_num = 1; row_num <= n; row_num++) { ` ` `  `            ``int` `count = 1; ` `            ``int` `c = 1; ` ` `  `            ``// Loop to generate each element of ith row ` `            ``for` `(``int` `i = 1; i <= row_num; i++) { ` ` `  `                ``c = c * (row_num - i) / i; ` ` `  `                ``// if c is odd ` `                ``// increment count ` `                ``if` `(c % 2 == 1) ` `                    ``count++; ` `            ``} ` ` `  `            ``// print count of odd elements ` `            ``Console.Write(count + ``" "``); ` `        ``} ` `    ``} ` ` `  `    ``// Driver code ` `    ``public` `static` `void` `Main() ` `    ``{ ` ` `  `        ``// Get n ` `        ``int` `n = 16; ` ` `  `        ``// Function call ` `        ``gouldSequence(n); ` `    ``} ` `} `

## PHP

 ` `

Output

```1 2 2 4 2 4 4 8 2 4 4 8 4 8 8 16
```

An Efficient Solution is based on the fact that the count of odd numbers in ith row of pascal’s Triangle is 2 raised to the count of 1’s in binary representation of i.

For Example

```for row=5
5 in binary = 101
count of 1's =2
22= 4

So, 5th row of pascal triangle will have 4 odd number

```

By counting 1’s in binary representation of every row number up to n, we can generate Gould’s Sequence up to n.

Below is the implementation of above idea-

## C++

 `// CPP program to generate ` `// Gould's Sequence ` ` `  `#include ` `using` `namespace` `std; ` ` `  `// Utility function to count odd numbers ` `// in ith row of Pascals's triangle ` `int` `countOddNumber(``int` `row_num) ` `{ ` ` `  `    ``// Count set bits in row_num ` ` `  `    ``// Initialize count as zero ` `    ``unsigned ``int` `count = 0; ` `    ``while` `(row_num) { ` `        ``count += row_num & 1; ` `        ``row_num >>= 1; ` `    ``} ` ` `  `    ``// Return 2^count ` `    ``return` `(1 << count); ` `} ` ` `  `// Function to generate gould's Sequence ` `void` `gouldSequence(``int` `n) ` `{ ` `    ``// loop to generate gould's Sequence up to n ` `    ``for` `(``int` `row_num = 0; row_num < n; row_num++) { ` ` `  `        ``cout << countOddNumber(row_num) << ``" "``; ` `    ``} ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` ` `  `    ``// Get n ` `    ``int` `n = 16; ` ` `  `    ``// Function call ` `    ``gouldSequence(n); ` ` `  `    ``return` `0; ` `} `

## Java

 `// JAVA program to generate ` `// Gould's Sequence ` ` `  `class` `GFG { ` ` `  `    ``// Utility function to count odd numbers ` `    ``// in ith row of Pascals's triangle ` `    ``static` `int` `countOddNumber(``int` `row_num) ` `    ``{ ` ` `  `        ``// Count set bits in row_num ` ` `  `        ``// Initialize count as zero ` `        ``int` `count = ``0``; ` `        ``while` `(row_num > ``0``) { ` `            ``count += row_num & ``1``; ` `            ``row_num >>= ``1``; ` `        ``} ` ` `  `        ``// Return 2^count ` `        ``return` `(``1` `<< count); ` `    ``} ` ` `  `    ``// Function to generate gould's Sequence ` `    ``static` `void` `gouldSequence(``int` `n) ` `    ``{ ` `        ``// loop to generate gould's Sequence up to n ` `        ``for` `(``int` `row_num = ``0``; row_num < n; row_num++) { ` ` `  `            ``System.out.print(countOddNumber(row_num) + ``" "``); ` `        ``} ` `    ``} ` ` `  `    ``// Driver code ` `    ``public` `static` `void` `main(String[] args) ` `    ``{ ` ` `  `        ``// Get n ` `        ``int` `n = ``16``; ` ` `  `        ``// Function call ` `        ``gouldSequence(n); ` `    ``} ` `} `

## Python3

 `# Python3 program to generate  ` `# Gould's Sequence  ` ` `  `# Utility function to count odd numbers  ` `# in ith row of Pascals's triangle  ` `def` `countOddNumber(row_num):  ` ` `  `    ``# Count set bits in row_num  ` `    ``# Initialize count as zero  ` `    ``count ``=` `0` `    ``while` `row_num !``=` `0``:  ` `        ``count ``+``=` `row_num & ``1` `        ``row_num >>``=` `1` ` `  `    ``# Return 2^count  ` `    ``return` `(``1` `<< count)  ` ` `  `# Function to generate gould's Sequence  ` `def` `gouldSequence(n):  ` ` `  `    ``# loop to generate gould's ` `    ``# Sequence up to n  ` `    ``for` `row_num ``in` `range``(``0``, n):  ` `        ``print``(countOddNumber(row_num), end ``=` `" "``)  ` ` `  `# Driver code  ` `if` `__name__ ``=``=` `"__main__"``: ` ` `  `    ``# Get n  ` `    ``n ``=` `16` ` `  `    ``# Function call  ` `    ``gouldSequence(n)  ` ` `  `# This code is contributed ` `# by Rituraj Jain `

## C#

 `// C# program to generate ` `// Gould's Sequence ` ` `  `using` `System; ` `class` `GFG { ` ` `  `    ``// Utility function to count odd numbers ` `    ``// in ith row of Pascals's triangle ` `    ``static` `int` `countOddNumber(``int` `row_num) ` `    ``{ ` ` `  `        ``// Count set bits in row_num ` ` `  `        ``// Initialize count as zero ` `        ``int` `count = 0; ` `        ``while` `(row_num > 0) { ` `            ``count += row_num & 1; ` `            ``row_num >>= 1; ` `        ``} ` ` `  `        ``// Return 2^count ` `        ``return` `(1 << count); ` `    ``} ` ` `  `    ``// Function to generate gould's Sequence ` `    ``static` `void` `gouldSequence(``int` `n) ` `    ``{ ` `        ``// loop to generate gould's Sequence up to n ` `        ``for` `(``int` `row_num = 0; row_num < n; row_num++) { ` ` `  `            ``Console.Write(countOddNumber(row_num) + ``" "``); ` `        ``} ` `    ``} ` ` `  `    ``// Driver code ` `    ``public` `static` `void` `Main() ` `    ``{ ` `        ``// Get n ` `        ``int` `n = 16; ` ` `  `        ``// Function call ` `        ``gouldSequence(n); ` `    ``} ` `} `

## PHP

 `>= 1; ` `    ``} ` ` `  `    ``// Return 2^count ` `    ``return` `(1 << ``\$count``); ` `} ` ` `  `// Function to generate gould's Sequence ` `function` `gouldSequence(``\$n``) ` `{ ` `    ``// loop to generate gould's Sequence up to n ` `    ``for` `(``\$row_num` `= 0;  ` `         ``\$row_num` `< ``\$n``; ``\$row_num``++) ` `    ``{ ` ` `  `        ``echo` `countOddNumber(``\$row_num``), ``" "``; ` `    ``} ` `} ` ` `  `// Driver code ` ` `  `// Get n ` `\$n` `= 16; ` ` `  `// Function call ` `gouldSequence(``\$n``); ` ` `  `// This code is contributed  ` `// by Sach_Code ` `?> `

Output

```1 2 2 4 2 4 4 8 2 4 4 8 4 8 8 16
```

A Better Solution ( Using Dynamic programming ) is based on the observation that after every power of 2 earlier terms got double up.

For Example

```
first term of the sequence is - 1
Now After every power of 2 we will double the value of previous terms

Terms up to 21  1 2
Terms up to 22  1 2 2 4
Terms up to 23  1 2 2 4 2 4 4 8
Terms up to 24  1 2 2 4 2 4 4 8 2 4 4 8 4 8 8 16

```

So, We can compute Gould’s Sequence terms after 2i by doubling the value of previous terms

Below is the implementation of above approach-

## C++

 `// CPP program to generate ` `// Gould's Sequence ` ` `  `#include ` `using` `namespace` `std; ` ` `  `// 32768 = 2^15 ` `#define MAX 32768 ` ` `  `// Array to store Sequence up to ` `// 2^16 = 65536 ` `int` `arr[2 * MAX]; ` ` `  `// Utility function to pre-compute odd numbers ` `// in ith row of Pascals's triangle ` `int` `gouldSequence() ` `{ ` ` `  `    ``// First term of the Sequence is 1 ` `    ``arr = 1; ` ` `  `    ``// Initilize i to 1 ` `    ``int` `i = 1; ` ` `  `    ``// Initilize p to 1 (i.e 2^i) ` `    ``// in each iteration ` `    ``// i will be pth power of 2 ` `    ``int` `p = 1; ` ` `  `    ``// loop to generate gould's Sequence ` `    ``while` `(i <= MAX) { ` ` `  `        ``// i is pth power of 2 ` `        ``// traverse the array ` `        ``// from j=0 to i i.e (2^p) ` ` `  `        ``int` `j = 0; ` ` `  `        ``while` `(j < i) { ` ` `  `            ``// double the value of arr[j] ` `            ``// and store to arr[i+j] ` `            ``arr[i + j] = 2 * arr[j]; ` `            ``j++; ` `        ``} ` ` `  `        ``// upadate i to next power of 2 ` `        ``i = (1 << p); ` ` `  `        ``// increment p ` `        ``p++; ` `    ``} ` `} ` ` `  `// Function to print gould's Sequence ` `void` `printSequence(``int` `n) ` `{ ` `    ``// loop to generate gould's Sequence up to n ` ` `  `    ``for` `(``int` `i = 0; i < n; i++) { ` `        ``cout << arr[i] << ``" "``; ` `    ``} ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` ` `  `    ``gouldSequence(); ` ` `  `    ``// Get n ` `    ``int` `n = 16; ` ` `  `    ``// Function call ` `    ``printSequence(n); ` ` `  `    ``return` `0; ` `} `

## Java

 `// JAVA program to generate ` `// Gould's Sequence ` ` `  `class` `GFG { ` ` `  `    ``// 32768 = 2^15 ` `    ``static` `final` `int` `MAX = ``32768``; ` ` `  `    ``// Array to store Sequence up to ` `    ``// 2^16 = 65536 ` `    ``static` `int``[] arr = ``new` `int``[``2` `* MAX]; ` ` `  `    ``// Utility function to pre-compute odd numbers ` `    ``// in ith row of Pascals's triangle ` `    ``static` `void` `gouldSequence() ` `    ``{ ` ` `  `        ``// First term of the Sequence is 1 ` `        ``arr[``0``] = ``1``; ` ` `  `        ``// Initilize i to 1 ` `        ``int` `i = ``1``; ` ` `  `        ``// Initilize p to 1 (i.e 2^i) ` `        ``// in each iteration ` `        ``// i will be pth power of 2 ` `        ``int` `p = ``1``; ` ` `  `        ``// loop to generate gould's Sequence ` `        ``while` `(i <= MAX) { ` ` `  `            ``// i is pth power of 2 ` `            ``// traverse the array ` `            ``// from j=0 to i i.e (2^p) ` ` `  `            ``int` `j = ``0``; ` ` `  `            ``while` `(j < i) { ` `                ``// double the value of arr[j] ` `                ``// and store to arr[i+j] ` `                ``arr[i + j] = ``2` `* arr[j]; ` `                ``j++; ` `            ``} ` ` `  `            ``// upadate i to next power of 2 ` `            ``i = (``1` `<< p); ` ` `  `            ``// increment p ` `            ``p++; ` `        ``} ` `    ``} ` ` `  `    ``// Function to print gould's Sequence ` `    ``static` `void` `printSequence(``int` `n) ` `    ``{ ` `        ``// loop to generate gould's Sequence up to n ` ` `  `        ``for` `(``int` `i = ``0``; i < n; i++) { ` `            ``System.out.print(arr[i] + ``" "``); ` `        ``} ` `    ``} ` ` `  `    ``// Driver code ` `    ``public` `static` `void` `main(String[] args) ` `    ``{ ` `        ``gouldSequence(); ` ` `  `        ``// Get n ` `        ``int` `n = ``16``; ` ` `  `        ``// Function call ` `        ``printSequence(n); ` `    ``} ` `} `

## Python3

 `# Python3 program to generate  ` `# Gould's Sequence  ` ` `  `# 32768 = 2^15  ` `MAX` `=` `32768` ` `  `# Array to store Sequence up to  ` `# 2^16 = 65536  ` `arr ``=` `[``None``] ``*` `(``2` `*` `MAX``) ` ` `  `# Utility function to pre-compute  ` `# odd numbers in ith row of Pascals's  ` `# triangle  ` `def` `gouldSequence():  ` ` `  `    ``# First term of the Sequence is 1  ` `    ``arr[``0``] ``=` `1` ` `  `    ``# Initilize i to 1  ` `    ``i ``=` `1` ` `  `    ``# Initilize p to 1 (i.e 2^i)  ` `    ``# in each iteration  ` `    ``# i will be pth power of 2  ` `    ``p ``=` `1` ` `  `    ``# loop to generate gould's Sequence  ` `    ``while` `i <``=` `MAX``:  ` ` `  `        ``# i is pth power of 2  ` `        ``# traverse the array  ` `        ``# from j=0 to i i.e (2^p)  ` `        ``j ``=` `0` ` `  `        ``while` `j < i:  ` ` `  `            ``# double the value of arr[j]  ` `            ``# and store to arr[i+j]  ` `            ``arr[i ``+` `j] ``=` `2` `*` `arr[j]  ` `            ``j ``+``=` `1` `         `  `        ``# upadate i to next power of 2  ` `        ``i ``=` `(``1` `<< p)  ` ` `  `        ``# increment p  ` `        ``p ``+``=` `1` `     `  `# Function to print gould's Sequence  ` `def` `printSequence(n):  ` ` `  `    ``# loop to generate gould's Sequence  ` `    ``# up to n  ` `    ``for` `i ``in` `range``(``0``, n): ` `        ``print``(arr[i], end ``=` `" "``)  ` `     `  `# Driver code  ` `if` `__name__ ``=``=` `"__main__"``:  ` ` `  `    ``gouldSequence()  ` ` `  `    ``# Get n  ` `    ``n ``=` `16` ` `  `    ``# Function call  ` `    ``printSequence(n)  ` ` `  `# This code is contributed ` `# by Rituraj Jain `

## C#

 `// C# program to generate ` `// Gould's Sequence ` ` `  `using` `System; ` `class` `GFG { ` ` `  `    ``// 32768 = 2^15 ` `    ``static` `int` `MAX = 32768; ` ` `  `    ``// Array to store Sequence up to ` `    ``// 2^16 = 65536 ` `    ``static` `int``[] arr = ``new` `int``[2 * MAX]; ` ` `  `    ``// Utility function to pre-compute odd numbers ` `    ``// in ith row of Pascals's triangle ` `    ``static` `void` `gouldSequence() ` `    ``{ ` ` `  `        ``// First term of the Sequence is 1 ` `        ``arr = 1; ` ` `  `        ``// Initialize i to 1 ` `        ``int` `i = 1; ` ` `  `        ``// Initialize p to 1 (i.e 2^i) ` `        ``// in each iteration ` `        ``// i will be pth power of 2 ` `        ``int` `p = 1; ` ` `  `        ``// loop to generate gould's Sequence ` `        ``while` `(i <= MAX) { ` ` `  `            ``// i is pth power of 2 ` `            ``// traverse the array ` `            ``// from j=0 to i i.e (2^p) ` ` `  `            ``int` `j = 0; ` ` `  `            ``while` `(j < i) { ` `                ``// double the value of arr[j] ` `                ``// and store to arr[i+j] ` `                ``arr[i + j] = 2 * arr[j]; ` `                ``j++; ` `            ``} ` ` `  `            ``// upadate i to next power of 2 ` `            ``i = (1 << p); ` ` `  `            ``// increment p ` `            ``p++; ` `        ``} ` `    ``} ` ` `  `    ``// Function to print gould's Sequence ` `    ``static` `void` `printSequence(``int` `n) ` `    ``{ ` `        ``// loop to generate gould's Sequence up to n ` ` `  `        ``for` `(``int` `i = 0; i < n; i++) { ` `            ``Console.Write(arr[i] + ``" "``); ` `        ``} ` `    ``} ` ` `  `    ``// Driver code ` `    ``public` `static` `void` `Main() ` `    ``{ ` ` `  `        ``gouldSequence(); ` ` `  `        ``// Get n ` `        ``int` `n = 16; ` ` `  `        ``// Function call ` `        ``printSequence(n); ` `    ``} ` `} `

## PHP

 ` `

Output

```1 2 2 4 2 4 4 8 2 4 4 8 4 8 8 16
```

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