所以我试图找到河内塔问题的封闭式解决方案。我知道递归关系是T（n）= 2T（n-1）+ 1，因为需要T（n-1）来回移动塔顶，这就是为什么有两个，并且＆ ＃34; + 1＆＃34;是移动基地。但是，我无法理解为什么封闭解决方案是2 ^ n - 1。
当我试图解决答案并且我使用回替换时，我得到：T（n）= 8T（n-3）+ 4 + 2 + 1，即T（n）= 2 ^ k（T（n-k））+ 2 ^ k-1 + 2 ^ k-2 + 2 ^ k-3其中k是台阶？我知道最后一部分也是几何系列，这意味着它是2 ^（n + 1） - 1 /（2-1）。但我无法理解答案的来源。
编辑： 是因为几何系列部分不是2 ^ k + 2 ^ k-1 + ... + 2 ^ k-k？这意味着几何级数不是2 ^ n + 1 - 1，而是2 ^ n - 1.并且我们使用H（0）作为基本情况 - ＆gt;所以H（n - k），使用k = n？
答案 0 :(得分：1)
您可以通过归纳轻松证明这一点。让我们假设n对于给定的T(n+1) = 2*T(n) + 1
= 2*(2^n-1) + 1
= 2^(n+1) - 2 + 1
= 2^(n+1) - 1
是正确的。然后：
T(0) = 0 = 2^0 - 1我们知道n它证明对于任何T(n) = 2^n - 1，等式<?xml version='1.0' encoding='UTF-8' ?>
<beans xmlns="http://www.springframework.org/schema/beans"
xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
xmlns:p="http://www.springframework.org/schema/p"
xmlns:aop="http://www.springframework.org/schema/aop"
xmlns:context="http://www.springframework.org/schema/context"
xmlns:tx="http://www.springframework.org/schema/tx"
xmlns:mvc="http://www.springframework.org/schema/mvc"
xmlns:oauth="http://www.springframework.org/schema/security/oauth2"
xmlns:security="http://www.springframework.org/schema/security"
xsi:schemaLocation="http://www.springframework.org/schema/beans http://www.springframework.org/schema/beans/spring-beans-4.2.xsd
http://www.springframework.org/schema/aop http://www.springframework.org/schema/aop/spring-aop-4.2.xsd
http://www.springframework.org/schema/context http://www.springframework.org/schema/context/spring-context-4.2.xsd
http://www.springframework.org/schema/tx http://www.springframework.org/schema/tx/spring-tx-4.2.xsd
http://www.springframework.org/schema/security http://www.springframework.org/schema/security/spring-security.xsd
http://www.springframework.org/schema/security/oauth2 http://www.springframework.org/schema/security/spring-security-oauth2-2.0.xsd
http://www.springframework.org/schema/mvc http://www.springframework.org/schema/mvc/spring-mvc-4.2.xsd">
<!--bean id="clientDetailServices" class="org.zinzu.mv.oauth2.ZZClientDetailServices"/-->
<http pattern="/oauth/token" create-session="stateless"
authentication-manager-ref="clientAuthenticationManager"
xmlns="http://www.springframework.org/schema/security">
<intercept-url pattern="/oauth/token" access="IS_AUTHENTICATED_FULLY" />
<anonymous enabled="false" />
<http-basic entry-point-ref="clientAuthenticationEntryPoint" />
<!-- include this only if you need to authenticate clients via request
parameters -->
<custom-filter ref="clientCredentialsTokenEndpointFilter"
after="BASIC_AUTH_FILTER" />
<access-denied-handler ref="oauthAccessDeniedHandler" />
</http>
<!-- This is where we tells spring security what URL should be protected
and what roles have access to them -->
<security:http pattern="/api/**" create-session="never" use-expressions="true"
entry-point-ref="oauthAuthenticationEntryPoint"
access-decision-manager-ref="accessDecisionManager">
<security:anonymous enabled="false" />
<security:intercept-url pattern="/api/**" access="hasRole('ROLE_USER')" />
<security:custom-filter ref="resourceServerFilter" before="PRE_AUTH_FILTER" />
<security:access-denied-handler ref="oauthAccessDeniedHandler" />
</security:http>
<bean id="oauthAuthenticationEntryPoint"
class="org.springframework.security.oauth2.provider.error.OAuth2AuthenticationEntryPoint">
<property name="realmName" value="test" />
</bean>
<bean id="clientAuthenticationEntryPoint"
class="org.springframework.security.oauth2.provider.error.OAuth2AuthenticationEntryPoint">
<property name="realmName" value="test/client" />
<property name="typeName" value="Basic" />
</bean>
<bean id="oauthAccessDeniedHandler"
class="org.springframework.security.oauth2.provider.error.OAuth2AccessDeniedHandler" />
<bean id="clientCredentialsTokenEndpointFilter"
class="org.springframework.security.oauth2.provider.client.ClientCredentialsTokenEndpointFilter">
<property name="authenticationManager" ref="clientAuthenticationManager" />
</bean>
<bean id="accessDecisionManager" class="org.springframework.security.access.vote.UnanimousBased"
xmlns="http://www.springframework.org/schema/beans">
<constructor-arg>
<list>
<bean class="org.springframework.security.oauth2.provider.vote.ScopeVoter" />
<bean class="org.springframework.security.access.vote.RoleVoter" />
<bean class="org.springframework.security.access.vote.AuthenticatedVoter" />
</list>
</constructor-arg>
</bean>
<authentication-manager id="clientAuthenticationManager"
xmlns="http://www.springframework.org/schema/security">
<authentication-provider user-service-ref="clientDetailsUserService" />
</authentication-manager>
<!-- This is simple authentication manager, with a hardcoded user/password
combination. We can replace this with a user defined service to get few users
credentials from DB -->
<authentication-manager alias="authenticationManager"
xmlns="http://www.springframework.org/schema/security">
<authentication-provider>
<user-service>
<user name="beingjavaguys" password="spring@java" authorities="ROLE_CLIENT" />
</user-service>
</authentication-provider>
</authentication-manager>
<bean id="clientDetailsUserService"
class="org.springframework.security.oauth2.provider.client.ClientDetailsUserDetailsService">
<constructor-arg ref="clientDetails" />
</bean>
<!-- This defined token store, we have used inmemory tokenstore for now
but this can be changed to a user defined one -->
<bean id="tokenStore"
class="org.springframework.security.oauth2.provider.token.store.InMemoryTokenStore" />
<bean id="approvalStore"
class="org.springframework.security.oauth2.provider.approval.TokenApprovalStore">
<property name="tokenStore" ref="tokenStore" />
</bean>
<!-- This is where we defined token based configurations, token validity
and other things -->
<bean id="tokenServices"
class="org.springframework.security.oauth2.provider.token.DefaultTokenServices">
<property name="tokenStore" ref="tokenStore" />
<property name="supportRefreshToken" value="true" />
<property name="accessTokenValiditySeconds" value="120" />
<property name="clientDetailsService" ref="clientDetails" />
</bean>
<bean id="requestFactory"
class="org.springframework.security.oauth2.provider.request.DefaultOAuth2RequestFactory">
<constructor-arg name="clientDetailsService" ref="clientDetails" />
</bean>
<bean id="userApprovalHandler"
class="org.springframework.security.oauth2.provider.approval.ApprovalStoreUserApprovalHandler">
<property name="clientDetailsService" ref="clientDetails"/>
<property name="approvalStore" ref="approvalStore"/>
<property name="requestFactory" ref="requestFactory"/>
</bean>
<oauth:authorization-server
client-details-service-ref="clientDetails" token-services-ref="tokenServices"
user-approval-handler-ref="userApprovalHandler">
<oauth:authorization-code />
<oauth:implicit />
<oauth:refresh-token />
<oauth:client-credentials />
<oauth:password />
</oauth:authorization-server>
<oauth:resource-server id="resourceServerFilter"
resource-id="test" token-services-ref="tokenServices" />
<oauth:client-details-service id="clientDetails">
<!-- client -->
<oauth:client client-id="restapp"
authorized-grant-types="authorization_code,client_credentials"
authorities="ROLE_CLIENT" scope="read,write,trust" secret="secret" />
<oauth:client client-id="restapp"
authorized-grant-types="password,authorization_code,refresh_token,implicit"
secret="restapp" authorities="ROLE_CLIENT" />
</oauth:client-details-service>
<security:global-method-security
pre-post-annotations="enabled" proxy-target-class="true">
<!--you could also wire in the expression handler up at the layer of the
http filters. See https://jira.springsource.org/browse/SEC-1452 -->
<security:expression-handler ref="oauthExpressionHandler" />
</security:global-method-security>
<oauth:expression-handler id="oauthExpressionHandler" />
<oauth:web-expression-handler id="oauthWebExpressionHandler" />
</beans>
都是正确的。
答案 1 :(得分：0)
有时工作的一个技巧是找到另一个关系更简单的函数。
所以我们从T(n)=2*T(n-1)+1开始。
它看起来类似于T(n)=2*T(n-1)，它有一个明显的解决方案。
因此，我们应该对公式进行转换，以便+1位于2*(...)内。
在这种情况下，它是T(n)+1=2*T(n-1)+2=2*(T(n-1)+1)。
所以T(n)+1=2^n*(T(0)+1)和T(n)=2^n*(T(0)+1)-1。