我觉得加深学习数学效果的方式，就是自己实现一遍，我这里用C语言实现了一下递归求麦粒总数的问题以求抛砖引玉
#include <stdio.h>
#include <stdlib.h>

struct result {
    long cur_grid_wheat;
    long sum_of_wheat;
};

struct result wheat(int grid)
{
    struct result rst;

    if (grid == 1) {
        rst.cur_grid_wheat = 1;
        rst.sum_of_wheat = 1;
    } else {
        struct result last_rst;
        last_rst = wheat(grid - 1);
        rst.cur_grid_wheat = last_rst.cur_grid_wheat * 2;
        rst.sum_of_wheat = last_rst.sum_of_wheat + rst.cur_grid_wheat;
    }

    return rst;
}

int main(int argc, char *argv[])
{
    struct result rst;
    int grid_index = 0;

    if (argc != 2) {
        printf("Usage: %s grid_index\n", argv[0]);
        return -1;
    }
    grid_index = atoi(argv[1]);

    rst = wheat(grid_index);

    printf("grid_index: %d\ncur_grid_wheat: %ld\nsum_of_wheat: %ld\n",
        grid_index, rst.cur_grid_wheat, rst.sum_of_wheat);

    return 0;
}

用Python实现数学归纳法，一开始使用math.pow()函数发现不对，Python中该函数会使用科学技术法导致结果出错
#-*- coding:UTF-8 -*-
class result(object):
    wheatNum = 0
    wheatTotalNum = 0

class getWheatTotalNum(object):
    '''
    函数说明：使用递归嵌套， 进行数学归纳法证明
    Param： k - 表示放到第几格 result - 表示当前格子的麦粒数
    Return: boolean - 放到第K格时是否成立
    '''
    def prove(self, k, result):
        if k == 1:
            if (2 ** 1 - 1) == 1:
                result.wheatNum = 1
                result.wheatTotalNum = 1
                return True
            else:
                return False

        else:
            proveOfPreviousOne = self.prove(k - 1, result)
            result.wheatNum *= 2
            result.wheatTotalNum += result.wheatNum
            proveOfCurrentOne = False
            if result.wheatTotalNum == (2 ** k - 1):
                proveOfCurrentOne = True
            if (proveOfPreviousOne & proveOfCurrentOne):
                return True
            else:
                return False


if __name__ == '__main__':
    grid = 64
    result = result()
    g = getWheatTotalNum()
    print(g.prove(grid, result))

<?php

class Result
{
    public $wheatNum = 0;
    public $wheatTotalNum = 0;
}

class Lession4_2
{

    public $result = NULL;

    public function __construct()
    {
        $this->result = new Result();
    }

    public function getResult()
    {
        return $this->result;
    }

    public function prove($n)
    {
        if ($n == 1) {
            if ((pow(2, 1) - 1) == 1) {
                $this->result->wheatNum = 1;
                $this->result->wheatTotalNum = 1;
                return true;
            } else {
                return false;
            }
        } else {
            $proveOfPreviousOne = $this->prove($n - 1);
            $this->result->wheatNum *= 2;
            $this->result->wheatTotalNum += $this->result->wheatNum;
            $proveOfCurrentOne = false;
            if ($this->result->wheatTotalNum == (pow(2, $n) - 1)) $proveOfCurrentOne = true;
            if ($proveOfCurrentOne && $proveOfPreviousOne) {
                return true;
            } else {
                return false;
            }
        }
    }
}

$a = new Lession4_2();
$result = $a->prove(13);
$result = $a->getResult();