See https://math.stackexchange.com/questions/1677748/why-do-ratios-of-these-fibonacci-type-sequences-approach-pi

estpi.c

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

#define ITERATIONS 85

typedef struct {
  uint64_t previous;
  uint64_t current;
} numpair;

int
main(void)
{
  numpair col_one = {.previous=12, .current=18};
  numpair col_two = {.previous=5, .current=5};

  int i;
  uint64_t tmp_num;
  for (i = 0; i < ITERATIONS; i++) {
    tmp_num = col_one.current;
    col_one.current = col_one.previous + col_one.current;
    col_one.previous = tmp_num;

    tmp_num = col_two.current;
    col_two.current = col_two.previous + col_two.current;
    col_two.previous = tmp_num;
  }

  printf("Pi ~= %f\n",
	((double)col_one.current/(double)col_two.current));
  return 0;
}