Stella Octangula Number in C++

Stella Octangula numbers are a set of numbers that possess some interesting geometric and number theory traits. The name "Stella Octangula" is Latin in origin, where "Stella" is the word for "Star", and "Octangula" indicates an octahedron, which is an eight-faced polyhedron. These numbers are obtained by applying the given formula as a formula to generate the sequence. In mathematics, a Stella Octangula number is a type of figurate number formed from the Stella Octangula (the shape of the form n(2n2 - 1)). The two Stella octangula numbers that are perfect squares are 1 (or one) and 9653449.

Example:

Approach 1:

The easiest way to perform this is to find the value of n(2n2 - 1) for starting from 0 and then compare this to x, i.e., the given number. We will continue increasing n and re-calculating n(2n2 - 1) until it becomes equal to x, which will indicate that x is a Stella octangula number. In case, the value of n is more than x, x should not be a Stella octangula number.

Example:

Let us take an example to illustrate the Stella Octangula Number in C++.

Output:

Yes

Explanation:

• In this example, the StellaOctangula() function accepts the key integer as an input and then checks whether it is a Stella Octangula number or not. It perform this by running a loop through the values of num and checking if the expression num*(2*num*num - 1) is equal to the key value. If it is the case, it returns true, and the key being a Stella Octangula number.
• In the main() function, a key variable is created and given the value of 14. The stellaOctangula() function is called using this key. After that, the program displays "Yes" if the function returns true and "No" if it returns false.

Approach 2:

It is another efficient approach to check whether the number is Stella Octangula or not. The method begins from n=1, and in each result iteration, the formula gets a new value: (2*n)^n^1. When we substitute for exponent, we get (2n)^ n^ -1 expression. Here we will speak about the X value and its corresponding expression value as z = 0.

We try to substitute the variable with the x at the next step. After that, we'll observe which variable is what x. In addition, the check is used if the condition is true, and to boolean type returns true when it is equal, and if not, go to binary search. We are focused on checking that the x and n^2-1 give the results of either true or false in the n/2 to n column and the equality.

Example 2:

Let us take another example to illustrate the Stella Octangula Number in C++.

Output:

Yes

Explanation:

• calculateExponent(num): This function computes the expression num * (2 * num * num - 1), which is a necessary condition for the numbers belonging to Stella Octangula.
• BinSearch(l, h, key): This function uses a binary search algorithm to find out the Stella Octangula numbers. It takes the limits of the lower and upper framing (L, H) and the key value (target) as input. If the calculated exponent is equal to the key, it is the targeted function. If true, it returns true, and if false, it returns false, signaling that the key is a Stella Octangula number. Otherwise, it changes the search area using comparisons between the computed exponent and the key.
• checkStellNumber(val): It is the core functionality that invokes a test of whether the integer val is a Stella Octangula number. At first, it puts away the first base case, which is when val equals 0 (which is Stella Octangula). In the next stage, it finds the starting value of the binary search by squaring the value of num until the calculated root is greater than the value of val. At last, it will call the BinSarch function with the equal range of [num/2, num] and the target value val, and it will get back with the results.

Conclusion:

In conclusion, it will show the differences in an approach to solving a difficult productively. The first algorithm presents a multiplication of n/linear speed to which it adds a linear expression -n^2 - n, and leads to (2*n) that is to be compared with a conventional identity of x. This initial method, which the author defined as "the slowest of all the methods of the check", is conceptualized by solving all problems in the task, and it means that the second algorithm is being done in more time. So, the run time by the power of factor n by two and by the n*(2nˆn-1)/n is multiplied. Therefore, in the next cycle, the two values are compared to x, which eliminates half of the search range. Thus, the search range is narrowed down. This is why the second algorithm is faster than the first one; the doubling of input size permits one to get the solution easily.