3. Series and Sequences

The namespace Sequences consists of the following functions

• fibonacci
• fibonacciList
• sumSeries
• sumMultiples
• factorial
• factorialList
• nextCollatz
• collatzLength
• champernowne

fibonacci

Finds the nth Fibonacci number using an iterative method. This utilizes much less space than the recursive procedure

Parameters:
int64_t n - The nth Fibonacci number to be found

Return value:
int64_t - The nth Fibonacci number

Sample usage (after including header file):

int64_t x = Sequences::fibonacci(1); //returns 1

fibonacciList

Returns an array of n Fibonacci numbers. Once again uses an iterative method to consume less memory. End is signalled by -1

Parameters:
int64_t n - The number of Fibonacci numbers needed

Return value:
int64_t* - A dynamically allocated array of size n + 1 having ith Fibonacci number at the ith index

Sample usage (after including header file):

int64_t* fib = Sequences::fibonacciList(11);
//Returns an array {0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, -1}

sumSeries

The sum of a series of n numbers each raised to coeff.

Parameters:
int n - The number of terms int coeff - The power each term is to be raised

Return value:
int64_t - The sum of first n natural numbers each raised to coeff

Sample usage (after including header file):

int64_t x = Sequences::sumSeries(3, 1); //returns 6
int64_t y = Sequences::sumSeries(3, 2); //returns 14
int64_t z = Sequences::sumSeries(3, 3); //returns 36

sumMultiples

Returns the sum of all multiples of multiple under the limit upperLimit. Runs in constant time.

Parameters:
int64_t upperLimit - The upper limit for summing the multiples int64_t multiple - The number whose multiples is to be summed

Return value:
int64_t - The sum of multiples of multiple under upperLimit

Sample usage (after including header file):

int64_t x = Sequences::sumMultiples(10, 2); //returns 20

factorial

Returns the factorial of a number. Uses an iterative method to conserve memory usage.

Parameters:
unsigned int n - The input number

Return value:
int64_t - The factorial of n

Sample usage (after including header file):

int64_t x = Sequences::factorial(0); //returns 1
int64_t y = Sequences::factorial(3); //returns 6

factorialList

Returns an array of numbers where ith index contains the factorial of i.

Parameters:
unsigned int n - The size of the array

Return value:
int64_t* - An array of size n + 1

Sample usage (after including header file):

int64_t* x = Sequences::factorialList(5);
//returns array {1, 1, 2, 6, 24, 120}

nextCollatz

Finds the next number in the collatz sequence

Parameters:
int64_t n - The previous number in the sequence

Return value:
int64_t - The number after n in the collatz sequence

Sample usage (after including header file):

int64_t x = Sequences::nextCollatz(5); // 16

collatzLength

Finds the length of the collatz sequence generated by starting at n

Parameters:
int n - The starting number of the sequence

Return value:
int64_t - The length of the collatz sequence generated by starting at n

Sample usage (after including header file):

int64_t x = Sequences::collatzLen(1); // 1

champernowne

Finds the champernowne constant with n digit precision

Parameters:
int64_t n - The length of the constant

Return value:
string - The constant with n digit precision

Sample usage (after including header file):

string champernowne = Sequences::champernowne(13); // "1234567891011"