[Mathematical-Tectonic matrix] Nefu 1113

according to test instructions, I have deduced the TN formula, Ti=ti.a+ti.b,ti.a=5*t (i-1). A+4*t ( i-1). B,ti.b=t (i-1). A+t (i-1). b

However, the following can not continue to push the formula SN!!!!

This problem is to find the first n of arbitrary sequence and, when the recurrence formula of SN is not obvious , the matrix is used to solve.

Set matrix a=, matrix f0=

Then set the Matrix s= (a+a2+a3 .... + an) *f0

The final answer is the sum of the two elements in the matrix S.

So how to beg a+a2+a3 .... + an?

You can continue to construct the following block matrix, where I is the unit matrix

Set r=, there are: r2=,r3=

Can be found in the upper right corner is i + A + a^2 + ... + a^n, one more I will be lost on the back.

The r^n can be obtained by fast power;

However, the above method for this problem is still tle, read the code discovery, can be deduced to further reduce the order of the Matrix, I here is the four-order, and the operation of the code is only three order.

Continue thinking:

See if we can directly deduce the formula of the S, see the explanation:

T[i] = dp[i][0]+dp[i][1]

=6*DP[I-1][1]+5*DP[I-1][0]

=6*T[I-1]-DP[I-1][0]

=6*t[i-1]-t[i-2]

According to S[i]=s[i-1]+t[i] can be calculated:

s[i]=s[i-1]+ 6*t[i-1]-t[i-2]

Then there are formulas:

Set r=, get it done!

Summary: The application of the matrix, carefully study the above structure matrix and derivation process, the first method of constructing the block matrix is very useful , it is not good for the SN formula directly constructs the matrix. But if the formula such as above s[i]=s[i-1]+ 6*t[i-1]-t[i-2] can deduce the recursive matrix of SN, it can reduce the complexity.

#include <stdio.h>#include<cstring>#include<cmath>#include<algorithm>#include<queue>using namespacestd;Const Long LongMoD =1000000007; structMa {Long Longm[3][3]; }; Maoperator*(ma A,ma b) {Ma C;  for(intI=0; i<3; i++)          for(intj=0; j<3; J + +) {C.m[i][j]=0;  for(intK =0; K <3; k++) {C.m[i][j]+ = (a.m[i][k]*B.m[k][j]); if(c.m[i][j]>=mod| | C.M[I][J]&LT;=-MOD) c.m[i][j]%=MoD; //C.m[i][j]%=mod;            }         }     returnC;} Ma mm[ $]; Long LongCalLong LongN) {intCur=0; Ma ans= {1,6,-1,              0,6,-1,              0,1,0             };  while(n) {if(n&1) {ans= ans*Mm[cur]; } cur++; N>>=1; }     Long Longtmp= A*ans.m[0][0]%mod+ **ans.m[0][1]%mod+6*ans.m[0][2]%mod+MoD; TMP%=MoD;  while(tmp<0) tmp+=MoD; printf ("%lld\n", TMP); returntmp;} //long Long ans[10000005];intMain () {Ma tmp= {1,6,-1,              0,6,-1,              0,1,0             }; mm[0] =tmp;  for(intI=1; i< -; i++) mm[i]=mm[i-1]*mm[i-1]; Long LongN;  while(SCANF ("%lld", &n)! =EOF) {         if(n==1) puts ("6"); Else if(n==2) puts (" A"); ElseCal (n3); }     return 0; }

[Mathematical-Tectonic matrix] Nefu 1113