P1028 数的计算-递推、递归

P1028 数的计算-递推、递归

题目传送门

本题看似传统的递归，由于数据可以到1000，所以传统递归会超时，实际上是一道递推题。本题解总共写了四个版本代码，分别是题意转换递归、升级版递归、打表代码、进阶版，打表和进阶版可以AC。

一、题意转换递归

根据题意简单转换了一个递归，遍历[1,n/2]，每个数字调用递归函数cal计算，提交超时。

#include 
using namespace std;

int n, result;

void cal(int a)
{
     
	result++;

	if(a>1){
     
		for(int j=1; j<=a/2; j++){
     
			cal(j);	
		}
	}

	if(a<=1)	return;
}
int main()
{
     	
	cin >> n;
	for(int i=1; i<=n/2; i++) cal(i);
	cout << result+1;
	return 0;	
} 

二、升级版递归

据说打表很好玩，于是准备整一个打表代码，在用小数据测试打表程序时发现：上面那个代码中的cal(n)恰好是结果，无需累加cal[1]~cal[n/2],虽然少了一些计算量，但还是一如既往超时。

#include 
using namespace std;

int n, result;

void cal(int a)
{
     
	result++;

	if(a>1){
     
		for(int j=1; j<=a/2; j++){
     
			cal(j);	
		}
	}

	if(a<=1)	return;
}
int main()
{
     	
	cin >> n;
	cal(n);
	cout << result;
	return 0;	
} 

三、打表代码

开始打表前首先要生成数据，思路就是把从0~1000的结果存在数组中，并把结果输出到本地文本内，直接复制。

下面是生成数据表的代码，由于0和1、2和3、3和4…两两结果一样，所以生成数据表时i每次迭代直接加2，可以节省一半时间：

#include 
#include  
using namespace std;

int n, result;

void cal(int a)
{
     
	result++;

	if(a>1){
     
		for(int j=1; j<=a/2; j++){
     
			cal(j);	
		}
	}

	if(a<=1)	return;
}
int main()
{
     	
	freopen("./table.txt", "w", stdout); 
	cout << "a[1002]={";
	for(int i=0; i<=1000; i+=2){
     
		result=0;
		cal(i);
		cout << result << "," << result << ",";
	}
	cout << "}";
	
	return 0;	
} 

下面是利用生成的数据表写的打表代码，很简单，可以AC，打表真香：

#include 
using namespace std;

int n, a[1001]={
     1,1,2,2,4,4,6,6,10,10,14,14,20,20,26,26,36,36,46,46,60,60,74,74,94,94,114,114,140,140,166,166,202,202,238,238,284,284,330,330,390,390,450,450,524,524,598,598,692,692,786,786,900,900,1014,1014,1154,1154,1294,1294,1460,1460,1626,1626,1828,1828,2030,2030,2268,2268,2506,2506,2790,2790,3074,3074,3404,3404,3734,3734,4124,4124,4514,4514,4964,4964,5414,5414,5938,5938,6462,6462,7060,7060,7658,7658,8350,8350,9042,9042,9828,9828,10614,10614,11514,11514,12414,12414,13428,13428,14442,14442,15596,15596,16750,16750,18044,18044,19338,19338,20798,20798,22258,22258,23884,23884,25510,25510,27338,27338,29166,29166,31196,31196,33226,33226,35494,35494,37762,37762,40268,40268,42774,42774,45564,45564,48354,48354,51428,51428,54502,54502,57906,57906,61310,61310,65044,65044,68778,68778,72902,72902,77026,77026,81540,81540,86054,86054,91018,91018,95982,95982,101396,101396,106810,106810,112748,112748,118686,118686,125148,125148,131610,131610,138670,138670,145730,145730,153388,153388,161046,161046,169396,169396,177746,177746,186788,186788,195830,195830,205658,205658,215486,215486,226100,226100,236714,236714,248228,248228,259742,259742,272156,272156,284570,284570,297998,297998,311426,311426,325868,325868,340310,340310,355906,355906,371502,371502,388252,388252,405002,405002,423046,423046,441090,441090,460428,460428,479766,479766,500564,500564,521362,521362,543620,543620,565878,565878,589762,589762,613646,613646,639156,639156,664666,664666,692004,692004,719342,719342,748508,748508,777674,777674,808870,808870,840066,840066,873292,873292,906518,906518,942012,942012,977506,977506,1015268,1015268,1053030,1053030,1093298,1093298,1133566,1133566,1176340,1176340,1219114,1219114,1264678,1264678,1310242,1310242,1358596,1358596,1406950,1406950,1458378,1458378,1509806,1509806,1564308,1564308,1618810,1618810,1676716,1676716,1734622,1734622,1795932,1795932,1857242,1857242,1922286,1922286,1987330,1987330,2056108,2056108,2124886,2124886,2197788,2197788,2270690,2270690,2347716,2347716,2424742,2424742,2506282,2506282,2587822,2587822,2673876,2673876,2759930,2759930,2850948,2850948,2941966,2941966,3037948,3037948,3133930,3133930,3235326,3235326,3336722,3336722,3443532,3443532,3550342,3550342,3663090,3663090,3775838,3775838,3894524,3894524,4013210,4013210,4138358,4138358,4263506,4263506,4395116,4395116,4526726,4526726,4665396,4665396,4804066,4804066,4949796,4949796,5095526,5095526,5248914,5248914,5402302,5402302,5563348,5563348,5724394,5724394,5893790,5893790,6063186,6063186,6240932,6240932,6418678,6418678,6605466,6605466,6792254,6792254,6988084,6988084,7183914,7183914,7389572,7389572,7595230,7595230,7810716,7810716,8026202,8026202,8252302,8252302,8478402,8478402,8715116,8715116,8951830,8951830,9200058,9200058,9448286,9448286,9708028,9708028,9967770,9967770,10239926,10239926,10512082,10512082,10796652,10796652,11081222,11081222,11379220,11379220,11677218,11677218,11988644,11988644,12300070,12300070,12625938,12625938,12951806,12951806,13292116,13292116,13632426,13632426,13988332,13988332,14344238,14344238,14715740,14715740,15087242,15087242,15475494,15475494,15863746,15863746,16268748,16268748,16673750,16673750,17096796,17096796,17519842,17519842,17960932,17960932,18402022,18402022,18862450,18862450,19322878,19322878,19802644,19802644,20282410,20282410,20782974,20782974,21283538,21283538,21804900,21804900,22326262,22326262,22869882,22869882,23413502,23413502,23979380,23979380,24545258,24545258,25135020,25135020,25724782,25724782,26338428,26338428,26952074,26952074,27591230,27591230,28230386,28230386,28895052,28895052,29559718,29559718,30251722,30251722,30943726,30943726,31663068,31663068,32382410,32382410,33130918,33130918,33879426,33879426,34657100,34657100,35434774,35434774,36243644,36243644,37052514,37052514,37892580,37892580,38732646,38732646,39605938,39605938,40479230,40479230,41385748,41385748,42292266,42292266,43234278,43234278,44176290,44176290,45153796,45153796,46131302,46131302,47146570,47146570,48161838,48161838,49214868,49214868,50267898,50267898,51361196,51361196,52454494,52454494,53588060,53588060,54721626,54721626,55897966,55897966,57074306,57074306,58293420,58293420,59512534,59512534,60777212,60777212,62041890,62041890,63352132,63352132,64662374,64662374,66020970,66020970,67379566,67379566,68786516,68786516,70193466,70193466,71651844,71651844,73110222,73110222,74620028,74620028,76129834,76129834,77694142,77694142,79258450,79258450,80877260,80877260,82496070,82496070,84172786,84172786,85849502,85849502,87584124,87584124,89318746,89318746,91114678,91114678,92910610,92910610,94767852,94767852,96625094,96625094,98547380,98547380,100469666,100469666,102456996,102456996,104444326,104444326,106500434,106500434,108556542,108556542,110681428,110681428,112806314,112806314,115004102,115004102,117201890,117201890,119472580,119472580,121743270,121743270,124090986,124090986,126438702,126438702,128863444,128863444,131288186,131288186,133794468,133794468,136300750,136300750,138888572,138888572,141476394,141476394,144150270,144150270,146824146,146824146,149584076,149584076,152344006,152344006,155194954,155194954,158045902,158045902,160987868,160987868,163929834,163929834,166967782,166967782,170005730,170005730,173139660,173139660,176273590,176273590,179508916,179508916,182744242,182744242,186080964,186080964,189417686,189417686,192861218,192861218,196304750,196304750,199855092,199855092,203405434,203405434,207068524,207068524,210731614,210731614,214507452,214507452,218283290,218283290,222177814,222177814,226072338,226072338,230085548,230085548,234098758,234098758,238237116,238237116,242375474,242375474,246638980,246638980,250902486,250902486,255297602,255297602,259692718,259692718,264219444,264219444,268746170,268746170,273411566,273411566,278076962,278076962,282881028,282881028,287685094,287685094,292634890,292634890,297584686,297584686,302680212,302680212,307775738,307775738,313024652,313024652,318273566,318273566,323675868,323675868,329078170,329078170,334641518,334641518,340204866,340204866,345929260,345929260,351653654,351653654,357547444,357547444,363441234,363441234,369504420,369504420,375567606,375567606,381808538,381808538,388049470,388049470,394468148,394468148,400886826,400886826,407492292,407492292,414097758,414097758,420890012,420890012,427682266,427682266,434670350,434670350,441658434,441658434,448842348,448842348,456026262,456026262,463415834,463415834,470805406,470805406,478400636,478400636,485995866,485995866,493806582,493806582,501617298,501617298,509643500,509643500,517669702,517669702,525922004,525922004,534174306,534174306,542652708,542652708,551131110,551131110,559846226,559846226,568561342,568561342,577513172,577513172,586465002,586465002,595665060,595665060,604865118,604865118,614313404,614313404,623761690,623761690,633469718,633469718,643177746,643177746,653145516,653145516,663113286,663113286,673353212,673353212,683593138,683593138,694105220,694105220,704617302,704617302,715413954,715413954,726210606,726210606,737291828,737291828,748373050,748373050,759752270,759752270,771131490,771131490,782808708,782808708,794485926,794485926,806474570,806474570,818463214,818463214,830763284,830763284,843063354,843063354,855689292,855689292,868315230,868315230,881267036,881267036,894218842,894218842,907510958,907510958,920803074,920803074,934435500,934435500,948067926,948067926,962056258,962056258,976044590,976044590,990388828,990388828,1004733066,1004733066,1019448806,1019448806,1034164546,1034164546,1049251788,1049251788,1064339030,1064339030,1079814524,1079814524,1095290018,1095290018,1111153764,1111153764,1127017510,1127017510,1143286258,1143286258,1159555006,1159555006,1176228756,1176228756,1192902506,1192902506,1209999302,1209999302,1227096098,1227096098,1244615940,1244615940,1262135782,1262135782,1280096714,1280096714,1298057646,1298057646,1316459668,1316459668,1334861690,1334861690,1353724140,1353724140,1372586590,1372586590,1391909468,1391909468,1411232346,1411232346,1431034990,1431034990,1450837634,1450837634,1471120044,1471120044,1491402454,1491402454,1512185428,1512185428,1532968402,1532968402,1554251940,1554251940,1575535478,1575535478,1597340378,1597340378,1619145278,1619145278,1641471540,1641471540,1663797802,1663797802,1686667684,1686667684,1709537566,1709537566,1732951068,1732951068,1756364570,1756364570,1780343950,1780343950,1804323330,1804323330,1828868588,1828868588,1853413846,1853413846,1878548866,1878548866,1903683886,1903683886,1929408668,1929408668,1955133450,1955133450,1981471878};
int main()
{
     	
	cin >> n; 
	cout << a[n] << endl;

	return 0;	
}

四、进阶版

打表太暴力了，还是要找一找规律才行，毕竟累加cal[1]~cal[n/2]居然和cal[n]一样，那肯定是有规律的。经过简单的分析，找到以下规律：

a[n]=s[n/2] //a[n]表示输入为n的答案

s[n]=s[n-1]+a[n] //s[n]表示a[]的前n项和

找到这就很简单了，整上下面两个初始条件：

a[0]=1, s[0]=1;

a[1]=1, s[1]=2;

我们就可以递推了:

a[2]=s[1]=2; s[2]=4;

a[3]=s[1]=2; s[3]=6;

a[4]=s[2]=4; s[4]=10;

a[5]=s[2]=4; s[5]=14;

a[6]=s[3]=6; s[6]=20;

我没用递归，直接用数组将中间结果保存了下来，可以AC。使用上面的递推规律用递归写应该也可以AC。上代码：

#include 
using namespace std;

int n, result, a[1001], s[1001];

int main()
{
     	
	cin >> n;
	a[0]=a[1]=1;
	s[0]=s[1]=2;
	
	for(int i=2; i<=n; i++){
     
		a[i]=s[i/2];
		s[i]=s[i-1]+a[i];	
	} 
	cout << a[n];
	return 0;	
} 

完毕！