@@ -52,6 +52,32 @@ T_4 = わ 1 + たす 1 + たす 2 = わ 4
5252
5353时间复杂度 $O(n),ドル空间复杂度 $O(1)$。其中 $n$ 为给定的整数。
5454
55+ ** 方法二:矩阵快速幂加速递推**
56+ 57+ 我们设 $Tib(n)$ 表示一个 1ドル \times 3$ 的矩阵 $\begin{bmatrix} T_n & T_ {n - 1} & T_ {n - 2} \end{bmatrix},ドル其中 $T_n,ドル $T_ {n - 1}$ 和 $T_ {n - 2}$ 分别表示第 $n$ 个、第 $n - 1$ 个和第 $n - 2$ 个泰波那契数。
58+ 59+ 我们希望根据 $Tib(n-1) = \begin{bmatrix} T_ {n - 1} & T_ {n - 2} & T_ {n - 3} \end{bmatrix}$ 推出 $Tib(n)$。也即是说,我们需要一个矩阵 $base,ドル使得 $Tib(n - 1) \times base = Tib(n),ドル即:
60+ 61+ $$
62+ \begin{bmatrix}
63+ T_{n - 1} & T_{n - 2} & T_{n - 3}
64+ \end{bmatrix} \times base = \begin{bmatrix} T_n & T_{n - 1} & T_{n - 2} \end{bmatrix}
65+ $$
66+ 67+ 由于 $T_n = T_ {n - 1} + T_ {n - 2} + T_ {n - 3},ドル所以矩阵 $base$ 为:
68+ 69+ $$
70+ \begin{bmatrix}
71+ 1 & 1 & 0 \\
72+ 1 & 0 & 1 \\
73+ 1 & 0 & 0
74+ \end{bmatrix}
75+ $$
76+ 77+ 我们定义初始矩阵 $res = \begin{bmatrix} 1 & 1 & 0 \end{bmatrix},ドル那么 $T_n$ 等于 $res$ 乘以 $base^{n - 3}$ 的结果矩阵中所有元素之和。使用矩阵快速幂求解即可。
78+ 79+ 时间复杂度 $O(\log n),ドル空间复杂度 $O(1)$。
80+ 5581<!-- tabs:start -->
5682
5783### ** Python3**
@@ -67,6 +93,35 @@ class Solution:
6793 return a
6894```
6995
96+ ``` python
97+ class Solution :
98+ def tribonacci (self n : int ) -> int :
99+ def mul (a : List[List[int ]], b : List[List[int ]]) -> List[List[int ]]:
100+ m, n = len (a), len (b[0 ])
101+ c = [[0 ] * n for _ in range (m)]
102+ for i in range (m):
103+ for j in range (n):
104+ for k in range (len (a[0 ])):
105+ c[i][j] = c[i][j] + a[i][k] * b[k][j]
106+ return c
107+ 108+ def pow (a : List[List[int ]], n : int ) -> List[List[int ]]:
109+ res = [[1 , 1 , 0 ]]
110+ while n:
111+ if n & 1 :
112+ res = mul(res, a)
113+ n >>= 1
114+ a = mul(a, a)
115+ return res
116+ 117+ if n == 0 :
118+ return 0
119+ if n < 3 :
120+ return 1
121+ a = [[1 , 1 , 0 ], [1 , 0 , 1 ], [1 , 0 , 0 ]]
122+ return sum (pow (a, n - 3 )[0 ])
123+ ```
124+ 70125### ** Java**
71126
72127<!-- 这里可写当前语言的特殊实现逻辑 -->
@@ -86,6 +141,51 @@ class Solution {
86141}
87142```
88143
144+ ``` java
145+ class Solution {
146+ public int tribonacci (int n ) {
147+ if (n == 0 ) {
148+ return 0 ;
149+ }
150+ if (n < 3 ) {
151+ return 1 ;
152+ }
153+ int [][] a = {{1 , 1 , 0 }, {1 , 0 , 1 }, {1 , 0 , 0 }};
154+ int [][] res = pow(a, n - 3 );
155+ int ans = 0 ;
156+ for (int x : res[0 ]) {
157+ ans += x;
158+ }
159+ return ans;
160+ }
161+ 162+ private int [][] mul (int [][] a , int [][] b ) {
163+ int m = a. length, n = b[0 ]. length;
164+ int [][] c = new int [m][n];
165+ for (int i = 0 ; i < m; ++ i) {
166+ for (int j = 0 ; j < n; ++ j) {
167+ for (int k = 0 ; k < b. length; ++ k) {
168+ c[i][j] += a[i][k] * b[k][j];
169+ }
170+ }
171+ }
172+ return c;
173+ }
174+ 175+ private int [][] pow (int [][] a , int n ) {
176+ int [][] res = {{1 , 1 , 0 }};
177+ while (n > 0 ) {
178+ if ((n & 1 ) == 1 ) {
179+ res = mul(res, a);
180+ }
181+ a = mul(a, a);
182+ n >> = 1 ;
183+ }
184+ return res;
185+ }
186+ }
187+ ```
188+ 89189### ** C++**
90190
91191``` cpp
@@ -104,6 +204,50 @@ public:
104204};
105205```
106206
207+ ```cpp
208+ class Solution {
209+ public:
210+ int tribonacci(int n) {
211+ if (n == 0) {
212+ return 0;
213+ }
214+ if (n < 3) {
215+ return 1;
216+ }
217+ vector<vector<ll>> a = {{1, 1, 0}, {1, 0, 1}, {1, 0, 0}};
218+ vector<vector<ll>> res = pow(a, n - 3);
219+ return accumulate(res[0].begin(), res[0].end(), 0);
220+ }
221+
222+ private:
223+ using ll = long long;
224+ vector<vector<ll>> mul(vector<vector<ll>>& a, vector<vector<ll>>& b) {
225+ int m = a.size(), n = b[0].size();
226+ vector<vector<ll>> c(m, vector<ll>(n));
227+ for (int i = 0; i < m; ++i) {
228+ for (int j = 0; j < n; ++j) {
229+ for (int k = 0; k < b.size(); ++k) {
230+ c[i][j] += a[i][k] * b[k][j];
231+ }
232+ }
233+ }
234+ return c;
235+ }
236+
237+ vector<vector<ll>> pow(vector<vector<ll>>& a, int n) {
238+ vector<vector<ll>> res = {{1, 1, 0}};
239+ while (n) {
240+ if (n & 1) {
241+ res = mul(res, a);
242+ }
243+ a = mul(a, a);
244+ n >>= 1;
245+ }
246+ return res;
247+ }
248+ };
249+ ```
250+ 107251### ** Go**
108252
109253``` go
@@ -116,6 +260,51 @@ func tribonacci(n int) int {
116260}
117261```
118262
263+ ``` go
264+ func tribonacci (n int ) (ans int ) {
265+ if n == 0 {
266+ return 0
267+ }
268+ if n < 3 {
269+ return 1
270+ }
271+ a := [][]int {{1 , 1 , 0 }, {1 , 0 , 1 }, {1 , 0 , 0 }}
272+ res := pow (a, n-3 )
273+ for _ , x := range res[0 ] {
274+ ans += x
275+ }
276+ return
277+ }
278+ 279+ func mul (a , b [][]int ) [][]int {
280+ m , n := len (a), len (b[0 ])
281+ c := make ([][]int , m)
282+ for i := range c {
283+ c[i] = make ([]int , n)
284+ }
285+ for i := 0 ; i < m; i++ {
286+ for j := 0 ; j < n; j++ {
287+ for k := 0 ; k < len (b); k++ {
288+ c[i][j] += a[i][k] * b[k][j]
289+ }
290+ }
291+ }
292+ return c
293+ }
294+ 295+ func pow (a [][]int , n int ) [][]int {
296+ res := [][]int {{1 , 1 , 0 }}
297+ for n > 0 {
298+ if n&1 == 1 {
299+ res = mul (res, a)
300+ }
301+ a = mul (a, a)
302+ n >>= 1
303+ }
304+ return res
305+ }
306+ ```
307+ 119308### ** JavaScript**
120309
121310``` js
@@ -137,6 +326,96 @@ var tribonacci = function (n) {
137326};
138327```
139328
329+ ``` js
330+ /**
331+ * @param {number} n
332+ * @return {number}
333+ */
334+ var tribonacci = function (n ) {
335+ if (n === 0 ) {
336+ return 0 ;
337+ }
338+ if (n < 3 ) {
339+ return 1 ;
340+ }
341+ const a = [
342+ [1 , 1 , 0 ],
343+ [1 , 0 , 1 ],
344+ [1 , 0 , 0 ],
345+ ];
346+ return pow (a, n - 3 )[0 ].reduce ((a , b ) => a + b);
347+ };
348+ 349+ function mul (a , b ) {
350+ const [m , n ] = [a .length , b[0 ].length ];
351+ const c = Array .from ({ length: m }, () => Array .from ({ length: n }, () => 0 ));
352+ for (let i = 0 ; i < m; ++ i) {
353+ for (let j = 0 ; j < n; ++ j) {
354+ for (let k = 0 ; k < b .length ; ++ k) {
355+ c[i][j] += a[i][k] * b[k][j];
356+ }
357+ }
358+ }
359+ return c;
360+ }
361+ 362+ function pow (a , n ) {
363+ let res = [[1 , 1 , 0 ]];
364+ while (n) {
365+ if (n & 1 ) {
366+ res = mul (res, a);
367+ }
368+ a = mul (a, a);
369+ n >>= 1 ;
370+ }
371+ return res;
372+ }
373+ ```
374+ 375+ ### ** TypeScript**
376+ 377+ ``` ts
378+ function tribonacci(n : number ): number {
379+ if (n === 0 ) {
380+ return 0 ;
381+ }
382+ if (n < 3 ) {
383+ return 1 ;
384+ }
385+ const = [
386+ [1 , 1 , 0 ],
387+ [1 , 0 , 1 ],
388+ [1 , 0 , 0 ],
389+ ];
390+ return pow (a , n - 3 )[0 ].reduce ((a , b ) => a + b );
391+ }
392+ 393+ function mul(a : number [][], b : number [][]): number [][] {
394+ const = [a .length , b [0 ].length ];
395+ const = Array .from ({ length: m }, () => Array .from ({ length: n }, () => 0 ));
396+ for (let i = 0 ; i < m ; ++ i ) {
397+ for (let j = 0 ; j < n ; ++ j ) {
398+ for (let k = 0 ; k < b .length ; ++ k ) {
399+ c [i ][j ] += a [i ][k ] * b [k ][j ];
400+ }
401+ }
402+ }
403+ return c ;
404+ }
405+ 406+ function pow(a : number [][], n : number ): number [][] {
407+ let res = [[1 , 1 , 0 ]];
408+ while (n ) {
409+ if (n & 1 ) {
410+ res = mul (res , a );
411+ }
412+ a = mul (a , a );
413+ n >>= 1 ;
414+ }
415+ return res ;
416+ }
417+ ```
418+ 140419### ** PHP**
141420
142421``` php
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