solving recursions with conditions
$begingroup$
I'd appreciate your help on solving the following recursions:
1) $a_n = 2^{-n} sum_{k=1}^n binom{n}{k}(a_k+a_{n-k})+ (0, text{if n even}) + (1, text{if n odd})$
2) $a_n = frac{1}{n}sum_{k=1}^n (a_{n-1} + a_{n-k}) + (1, text{if k = 1 or k = n})$
3) $a_n = sum_{k=0}^n binom{n}{k}(a_k + a_{n-k}) + (0, text{if n even}) + (1, text{if n odd})$
4) $a_n = 2^{-n} sum_{k=0}^n binom{n}{k}(a_k+a_{n-k})+ (0, text{if n even}) + (1, text{if n odd})$, whereby $a_0 = 1, a_1 = 0$.
I've tried to subtract $an$ from $a_{n+1}$ in order to get an easy recursion to solve, but in all of the recursions, the term was still too complicated. So did I miss something to make it easier or is there an other way to solve this kind of recursion?
Looking forward to your replies!
analysis recurrence-relations recursion
asked Jan 25 at 15:47
StudentuStudentu
1279
$endgroup$
add a comment |
$begingroup$
I'd appreciate your help on solving the following recursions:
1) $a_n = 2^{-n} sum_{k=1}^n binom{n}{k}(a_k+a_{n-k})+ (0, text{if n even}) + (1, text{if n odd})$
2) $a_n = frac{1}{n}sum_{k=1}^n (a_{n-1} + a_{n-k}) + (1, text{if k = 1 or k = n})$
3) $a_n = sum_{k=0}^n binom{n}{k}(a_k + a_{n-k}) + (0, text{if n even}) + (1, text{if n odd})$
4) $a_n = 2^{-n} sum_{k=0}^n binom{n}{k}(a_k+a_{n-k})+ (0, text{if n even}) + (1, text{if n odd})$, whereby $a_0 = 1, a_1 = 0$.
I've tried to subtract $an$ from $a_{n+1}$ in order to get an easy recursion to solve, but in all of the recursions, the term was still too complicated. So did I miss something to make it easier or is there an other way to solve this kind of recursion?
Looking forward to your replies!
analysis recurrence-relations recursion
asked Jan 25 at 15:47
StudentuStudentu
1279
$endgroup$
add a comment |
$begingroup$
I'd appreciate your help on solving the following recursions:
1) $a_n = 2^{-n} sum_{k=1}^n binom{n}{k}(a_k+a_{n-k})+ (0, text{if n even}) + (1, text{if n odd})$
2) $a_n = frac{1}{n}sum_{k=1}^n (a_{n-1} + a_{n-k}) + (1, text{if k = 1 or k = n})$
3) $a_n = sum_{k=0}^n binom{n}{k}(a_k + a_{n-k}) + (0, text{if n even}) + (1, text{if n odd})$
4) $a_n = 2^{-n} sum_{k=0}^n binom{n}{k}(a_k+a_{n-k})+ (0, text{if n even}) + (1, text{if n odd})$, whereby $a_0 = 1, a_1 = 0$.
I've tried to subtract $an$ from $a_{n+1}$ in order to get an easy recursion to solve, but in all of the recursions, the term was still too complicated. So did I miss something to make it easier or is there an other way to solve this kind of recursion?
Looking forward to your replies!
analysis recurrence-relations recursion
asked Jan 25 at 15:47
StudentuStudentu
1279
$endgroup$
I'd appreciate your help on solving the following recursions:
1) $a_n = 2^{-n} sum_{k=1}^n binom{n}{k}(a_k+a_{n-k})+ (0, text{if n even}) + (1, text{if n odd})$
2) $a_n = frac{1}{n}sum_{k=1}^n (a_{n-1} + a_{n-k}) + (1, text{if k = 1 or k = n})$
3) $a_n = sum_{k=0}^n binom{n}{k}(a_k + a_{n-k}) + (0, text{if n even}) + (1, text{if n odd})$
4) $a_n = 2^{-n} sum_{k=0}^n binom{n}{k}(a_k+a_{n-k})+ (0, text{if n even}) + (1, text{if n odd})$, whereby $a_0 = 1, a_1 = 0$.
I've tried to subtract $an$ from $a_{n+1}$ in order to get an easy recursion to solve, but in all of the recursions, the term was still too complicated. So did I miss something to make it easier or is there an other way to solve this kind of recursion?
Looking forward to your replies!
analysis recurrence-relations recursion
analysis recurrence-relations recursion
asked Jan 25 at 15:47
StudentuStudentu
1279
asked Jan 25 at 15:47
StudentuStudentu
1279
asked Jan 25 at 15:47
StudentuStudentu
1279
asked Jan 25 at 15:47
StudentuStudentu
1279
asked Jan 25 at 15:47
StudentuStudentu
1279
1279
add a comment |
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3087234%2fsolving-recursions-with-conditions%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3087234%2fsolving-recursions-with-conditions%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
