Step by Step solution about How to resolve the algorithm Motzkin numbers step by step in the C programming language

This C code explores the properties of the OEIS sequence A030452, defined as M(n) = (4^n + 4 - 2^(n+3)) / 6 for n ≥ 0, with an initial seed value of M(0) = M(1) = 1. The program checks if each value in the sequence is a prime number and prints the corresponding results.

Here's a step-by-step explanation of the code:

1. Header Inclusions:

   • The code includes several standard C library headers:
     • <locale.h> for locale settings related to number formatting.
     • <stdbool.h> for the bool data type.
     • <stdint.h> for the uint64_t data type (unsigned 64-bit integer).
     • <stdio.h> for input and output operations.

2. is_prime Function:

   • This function checks whether a given unsigned 64-bit integer n is a prime number.
   • It uses optimizations to quickly check for small prime factors (2 and 3) and then checks for other factors up to the square root of n.
   • If n is a prime number, the function returns true; otherwise, it returns false.

3. main Function:

   • Sets the locale to use the system's default settings for number formatting with setlocale(LC_ALL, "").
   • Prints a header line with column labels: "n", "M(n)", and "Prime?".
   • Initializes variables m0 and m1 with the initial seed values for the OEIS sequence.

4. Loop for Calculating M(n):

   • The loop iterates from i = 0 to i < 42.
   • For each iteration, it calculates the next value in the OEIS sequence as:
     • m = (m1 * (2 * i + 1) + m0 * (3 * i - 3)) / (i + 2) for i > 1.
     • m = 1 for i = 0 and i = 1 (the initial seed values).
   • The calculated value m is stored for printing.

5. Printing Results:

   • For each value of m, the program prints:
     • i: The current index in the sequence (starting from 0).
     • m: The calculated value of M(n) for the current index.
     • "true" or "false" depending on whether m is a prime number.

6. Updating Seed Values:

   • After printing the results for the current iteration, the program updates m0 to m1 and m1 to the newly calculated m for the next iteration of the loop.

#include <locale.h>
#include <stdbool.h>
#include <stdint.h>
#include <stdio.h>

bool is_prime(uint64_t n) {
    if (n < 2)
        return false;
    if (n % 2 == 0)
        return n == 2;
    if (n % 3 == 0)
        return n == 3;
    for (uint64_t p = 5; p * p <= n; p += 4) {
        if (n % p == 0)
            return false;
        p += 2;
        if (n % p == 0)
            return false;
    }
    return true;
}

int main() {
    setlocale(LC_ALL, "");
    printf(" n          M(n)             Prime?\n");
    printf("-----------------------------------\n");
    uint64_t m0 = 1, m1 = 1;
    for (uint64_t i = 0; i < 42; ++i) {
        uint64_t m =
            i > 1 ? (m1 * (2 * i + 1) + m0 * (3 * i - 3)) / (i + 2) : 1;
        printf("%2llu%'25llu  %s\n", i, m, is_prime(m) ? "true" : "false");
        m0 = m1;
        m1 = m;
    }
}