杭电1005Number Sequence
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Number Sequence
Time Limit: 2000/1000 MS (Java/Others) Memory Limit: 65536/32768 K (Java/Others)
Total Submission(s): 126495 Accepted Submission(s): 30758
Total Submission(s): 126495 Accepted Submission(s): 30758
Problem Description
A number sequence is defined as follows:
f(1) = 1, f(2) = 1, f(n) = (A * f(n - 1) + B * f(n - 2)) mod 7.
Given A, B, and n, you are to calculate the value of f(n).
f(1) = 1, f(2) = 1, f(n) = (A * f(n - 1) + B * f(n - 2)) mod 7.
Given A, B, and n, you are to calculate the value of f(n).
Input
The input consists of multiple test cases. Each test case contains 3 integers A, B and n on a single line (1 <= A, B <= 1000, 1 <= n <= 100,000,000). Three zeros signal the end of input and this test case is not to be processed.
Output
For each test case, print the value of f(n) on a single line.
Sample Input
1 1 31 2 100 0 0
Sample Output
25
#include<stdio.h>int main(){int n,x,y,z,i;int a[100];a[1]=1;a[2]=1;while(~scanf("%d %d %d",&x,&y,&z)){if(x==0&&y==0&&z==0)return 0;for(i=3;i<=49;i++) {a[i]=(x*a[i-1]+y*a[i-2])%7;}printf("%d\n",a[z%49]);}return 0;}注意:int的数值范围有限,当数值z大时会有数值溢出,又因为对7取余,所以以49为周期取z%49的数值代替z;
for循环要i<=49结束
根据周期,可以快速获得结果,且周期必定小于49.
周期的原因:如果<f(x),f(x+1)>的值与<f(x+T),f(x+1+T)>一致,那么以后的序列就完全一致了。
周期小于49的原因: 由于mod 7,所以<f(x),f(x+1)>的可能为49个,那么50个这样的<f(x),f(x+1)>必定有两个相同(鸽笼原理),所以最终可以确定周期小于49.
2、需要注意的是:并不是整个序列是有周期的,而是其中去掉头部一部分以后是有周期的!!!举个特殊的例子A=7,B=7,那么{f(x)}为{1,1,0,0,0,0,...},其在去掉头部的两个1后是有周期的,所以应该求出这个头部。类似的例子还有{A=2,B=7},{A=3,B=7}...
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