Phương pháp hàm sinh trong toán rời rạc

```

DISCRETE MATHEMATICS / GENERATING FUNCTIONS

Phương pháp hàm sinh trong toán rời rạc và các dãy số

a<sub>0</sub>, a<sub>1</sub>, a<sub>2</sub>, ...  →  A(x) = a<sub>0</sub> + a<sub>1</sub>x + a<sub>2</sub>x<sup>2</sup> + ···

Phương pháp hàm sinh (generating function) biến một dãy số thành một chuỗi lũy thừa để những quan hệ giữa các phần tử có thể được xử lý bằng đại số. Nếu dãy ban đầu là a<sub>0</sub>, a<sub>1</sub>, a<sub>2</sub>, ..., ta mã hóa nó thành một hàm A(x) sao cho hệ số của x<sup>n</sup> chính là a<sub>n</sub>.

```

```

Tóm tắt nội dung

Ý tưởng cốt lõi của hàm sinh là chuyển bài toán về dãy số thành bài toán đại số. Các phép dịch chỉ số như a<sub>n-1</sub> hoặc a<sub>n-2</sub> có thể được biểu diễn thông qua xA(x) và x<sup>2</sup>A(x). Nhờ đó, một hệ vô hạn các quan hệ giữa các phần tử có thể được rút gọn thành một phương trình theo A(x).

```

1. Hàm sinh biểu diễn dãy số như thế nào?

Cho dãy a₀, a₁, a₂, a₃, ... . Hàm sinh thông thường của dãy được định nghĩa bởi biểu thức:

A(x) = a₀ + a₁x + a₂x² + a₃x³ + ···

Ký hiệu [xⁿ]A(x) thường được dùng để chỉ hệ số của xⁿ trong A(x). Vì vậy:

[xⁿ]A(x) = a_n

Chẳng hạn, dãy 1, 1, 1, 1, ... có hàm sinh:

A(x) = 1 + x + x² + x³ + ··· = 1 1 − x

2. Dịch chỉ số và công thức truy hồi

Nếu A(x)=Σ_n≥0a_nxⁿ, thì nhân với x tạo ra:

xA(x) = a₀x + a₁x² + a₂x³ + ···

Nghĩa là hệ số được dịch sang phải một vị trí. Tương tự, x²A(x) dịch chúng hai vị trí. Chính cơ chế này cho phép các thành phần như a_n-1 và a_n-2 trong công thức truy hồi được chuyển thành phép nhân đại số.

3. Ví dụ: tìm hàm sinh của dãy Fibonacci

Dãy Fibonacci được xác định bởi F₀=0, F₁=1 và:

F_n = F_n-1 + F_n-2,   n ≥ 2

Đặt F(x)=Σ_n≥0F_nxⁿ . Sau khi nhân công thức truy hồi với xⁿ, lấy tổng từ n=2 và xử lý các điều kiện đầu, ta thu được:

F(x) − x = xF(x) + x²F(x)

Gom các thành phần chứa F(x) về một phía:

F(x)(1 − x − x²) = x

Do đó hàm sinh của dãy Fibonacci là:

F(x) = x 1 − x − x²

4. Minh họa bằng Python

Trong thực tế, ta thường chỉ cần một số hữu hạn hệ số đầu tiên. Đoạn mã sau nhân hai đa thức hữu hạn để mô phỏng phép nhân hai hàm sinh.

```

Python

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    var marker = document.getElementById('cs-ad-end');
    var article = document.querySelector('body.item-view .post-body.entry-content');
    if(!marker || !article || !marker.getAttribute('data-src')){ return; }
    var MIN_ENGAGED_MS = 8000;
    var MIN_ARTICLE_PROGRESS = 0.52;
    var READS_PER_REQUEST = 2;
    var MIN_REQUEST_GAP_MS = 150000;
    var READ_DEDUP_MS = 45 * 60000;
    var STORAGE_KEY = 'cs_monetag_reading_v3';
    var id = window.location.pathname.replace(/\/$/, '') || '/';
    var current = Date.now();
    var state = {readIds:[], recent:[], lastRequest:0};
    var requested = false, counted = false, pending = false;
    var visibleSince = document.visibilityState !== 'hidden' ? current : 0;
    var activeTimeMs = 0, lastCheck = 0, timer = null;
    try{
      var stored = JSON.parse(localStorage.getItem(STORAGE_KEY) || 'null');
      if(stored && Array.isArray(stored.readIds) && Array.isArray(stored.recent)){
        state = {readIds:stored.readIds.filter(function(v){ return typeof v === 'string'; }).slice(-6),
          recent:stored.recent.filter(function(v){ return v && typeof v.id === 'string' && Number.isFinite(v.at) && current - v.at >= 0 && current - v.at < READ_DEDUP_MS; }).slice(-20),
          lastRequest:Number(stored.lastRequest) || 0};
      }
    }catch(e){ /* Storage is optional; a private browser may use per-page behavior. */ }
    function save(){ try{ localStorage.setItem(STORAGE_KEY, JSON.stringify(state)); }catch(e){} }
    function readingProgress(){
      var rect = article.getBoundingClientRect();
      var height = Math.max(article.offsetHeight, rect.height, 1);
      return Math.max(0, Math.min(1, (window.innerHeight - rect.top) / height));
    }
    function engagedTime(){ return activeTimeMs + (visibleSince ? Date.now() - visibleSince : 0); }
    function requestAd(now){
      if(requested){ return; }
      requested = true;
      state.readIds = [];
      state.lastRequest = now;
      save();
      cleanup();
      var tag = document.createElement('script');
      tag.setAttribute('data-zone', marker.getAttribute('data-zone'));
      tag.setAttribute('data-cfasync', 'false');
      tag.async = true;
      tag.src = marker.getAttribute('data-src');
      (document.body || document.documentElement).appendChild(tag);
    }
    function check(){
      if(requested || document.visibilityState === 'hidden'){ return; }
      var now = Date.now();
      if(engagedTime() < MIN_ENGAGED_MS || readingProgress() < MIN_ARTICLE_PROGRESS){ return; }
      if(!counted){
        counted = true;
        if(!state.recent.some(function(v){ return v.id === id; })){
          state.readIds.push(id);
          state.readIds = state.readIds.slice(-6);
          state.recent.push({id:id, at:now});
          state.recent = state.recent.slice(-20);
          save();
        }
      }
      if(state.readIds.length >= READS_PER_REQUEST){
        if(now - state.lastRequest >= MIN_REQUEST_GAP_MS){ requestAd(now); }
        else if(!pending){
          pending = true;
          timer = setTimeout(function(){ pending = false; check(); }, Math.min(60000, Math.max(1000, MIN_REQUEST_GAP_MS - (now - state.lastRequest))));
        }
      }
    }
    function onScroll(){ var now = Date.now(); if(now - lastCheck >= 200){ lastCheck = now; check(); } }
    function onVisibility(){
      if(document.visibilityState === 'hidden'){
        if(visibleSince){ activeTimeMs += Date.now() - visibleSince; visibleSince = 0; }
      }else{ visibleSince = Date.now(); check(); }
    }
    function cleanup(){
      window.removeEventListener('scroll', onScroll);
      window.removeEventListener('resize', check);
      document.removeEventListener('visibilitychange', onVisibility);
      window.removeEventListener('pageshow', check);
      if(timer){ clearTimeout(timer); timer = null; }
    }
    window.addEventListener('scroll', onScroll, {passive:true});
    window.addEventListener('resize', check, {passive:true});
    document.addEventListener('visibilitychange', onVisibility);
    window.addEventListener('pageshow', check);
    timer = setTimeout(function(){ timer = null; check(); }, MIN_ENGAGED_MS + 100);
    check();
  });

  // Show dates on the Most Viewed cards as dd/mm/yyyy, matching the catalog cards.
  onReady(function formatPopularDates(){
    document.querySelectorAll('.homepage-popular time, .homepage-popular abbr.published').forEach(function(node){
      var date = new Date(node.getAttribute('datetime') || node.getAttribute('title'));
      if(Number.isFinite(date.getTime())){
        node.textContent = date.toLocaleDateString('vi-VN', {day:'2-digit', month:'2-digit', year:'numeric'});
      }
    });
  });

  onReady(function initHeaderSearch(){
    var header = document.querySelector('.centered-top-container .centered-top');
    var search = header && header.querySelector('.search');
    var toggle = search && search.querySelector('.search-expand');
    var input = search && search.querySelector('input[name="q"]');
    if(!toggle || !input){ return; }

    function setOpen(open, restoreFocus){
      search.classList.toggle('focused', open);
      header.classList.toggle('search-focused', open);
      toggle.setAttribute('aria-expanded', String(open));
      if(open){
        document.body.classList.remove('header-hidden');
        input.focus({preventScroll:true});
      }else if(restoreFocus){
        toggle.focus({preventScroll:true});
      }
    }
    // One toggle handler prevents Rockpool and the custom header from toggling twice.
    document.addEventListener('click', function(event){
      if(toggle.contains(event.target)){
        event.preventDefault();
        event.stopImmediatePropagation();
        setOpen(!search.classList.contains('focused'), false);
      }else if(!search.contains(event.target)){
        setOpen(false, false);
      }
    }, true);
    search.addEventListener('keydown', function(event){
      if(event.key === 'Escape'){
        event.preventDefault();
        event.stopPropagation();
        setOpen(false, true);
      }
    });
    new MutationObserver(function(){
      toggle.setAttribute('aria-expanded', String(search.classList.contains('focused')));
    }).observe(search, {attributes:true, attributeFilter:['class']});
  });

  // Progressive catalog enhancement: native HTML remains visible/indexable while
  // missing cards are filled from the feed. Blogger sometimes server-renders only
  // three posts because of its auto-pagination or large HTML-heavy articles.
  onReady(function fillShortBloggerCatalog(){
    var url = new URL(window.location.href);
    if(url.searchParams.has('cs-page') || url.searchParams.has('updated-max') ||
       url.searchParams.has('updated-min') || url.searchParams.has('start-index') ||
       url.searchParams.has('q') || !window.fetch || !window.AbortController ||
       !document.body.classList.contains('feed-view')){ return; }
    var labelMatch = url.pathname.match(/^\/search\/label\/([^/]+)\/?$/);
    if(!labelMatch && url.pathname !== '/' && url.pathname !== '/search'){ return; }
    var widget = document.querySelector('.widget.Blog');
    var grid = widget && widget.querySelector('.blog-posts');
    var pager = widget && widget.querySelector('#blog-pager');
    if(!grid || !pager){ return; }
    var label = '';
    try{ label = labelMatch ? decodeURIComponent(labelMatch[1]) : ''; }catch(e){ return; }
    var feedPath = '/feeds/posts/default' + (label ? '/-/' + encodeURIComponent(label) : '');
    var PAGE_SIZE = 12;
    var seen = new Set();
    function normalize(href){
      try{
        var a = new URL(href, window.location.href);
        return a.origin === url.origin ? a.pathname.replace(/\/$/, '') : '';
      }catch(e){ return ''; }
    }
    grid.querySelectorAll('.post-outer-container .post-title a[href]').forEach(function(link){
      var key = normalize(link.href); if(key){ seen.add(key); }
    });
    var nativeCount = seen.size;
    // Do not load a duplicate feed when Blogger has already rendered a full page.
    if(nativeCount >= PAGE_SIZE){ return; }
    var nextIndex = PAGE_SIZE + 1, total = 0, running = false;
    var control = document.createElement('button');
    control.type = 'button'; control.className = 'cs-load-more';
    control.textContent = 'Xem thêm bài viết';
    control.setAttribute('aria-label', 'Tải thêm 12 bài viết');
    var status = document.createElement('span');
    status.className = 'cs-feed-status'; status.setAttribute('role', 'status');
    pager.prepend(status); pager.prepend(control);
    control.hidden = true;
    var nativeMore = pager.querySelector('a.blog-pager-older-link');
    if(nativeMore){ nativeMore.setAttribute('aria-label', 'Xem trang bài đăng cũ trên Blogger'); }
    function plainText(html){
      var temp = document.createElement('template'); temp.innerHTML = html || '';
      temp.content.querySelectorAll('script,style,noscript,iframe,svg,object').forEach(function(e){ e.remove(); });
      return (temp.content.textContent || '').replace(/\s+/g, ' ').trim();
    }
    function createCard(entry, index){
      var link = (entry.link || []).find(function(a){ return a.rel === 'alternate' && a.type === 'text/html'; });
      var path = link && normalize(link.href);
      if(!path || seen.has(path)){ return null; }
      var title = plainText(entry.title && entry.title.$t) || 'Bài viết';
      var article = document.createElement('article'); article.className = 'post-outer-container cs-catalog-card';
      var outer = document.createElement('div'); outer.className = 'post-outer';
      var cover = document.createElement('a'); cover.className = 'snippet-thumbnail thumbnail-empty';
      cover.href = link.href; cover.setAttribute('aria-label', title);
      cover.setAttribute('data-post-id', (entry.id && entry.id.$t || '').split('.post-').pop());
      var thumb = postThumbnails.preview(entry);
      if(thumb){ postThumbnails.attach(cover, thumb, index < 3 ? 'visible' : false, function(){ return postThumbnails.preview(entry, true); }); }
      var content = document.createElement('div'); content.className = 'post-content';
      var heading = document.createElement('h3'); heading.className = 'post-title entry-title';
      var titleLink = document.createElement('a'); titleLink.href = link.href; titleLink.textContent = title;
      heading.appendChild(titleLink); content.appendChild(heading);
      var published = entry.published && new Date(entry.published.$t);
      if(published && Number.isFinite(published.getTime())){
        var meta = document.createElement('div'); meta.className = 'post-header-container';
        var time = document.createElement('time'); time.dateTime = published.toISOString();
        time.textContent = published.toLocaleDateString('vi-VN', {day:'2-digit',month:'2-digit',year:'numeric'});
        meta.appendChild(time); content.appendChild(meta);
      }
      var excerpt = plainText((entry.summary || entry.content || {}).$t);
      if(excerpt){
        var p = document.createElement('p'); p.className = 'post-body';
        p.textContent = excerpt.length > 210 ? excerpt.slice(0, 207) + '…' : excerpt;
        content.appendChild(p);
      }
      outer.append(cover, content); article.appendChild(outer);
      seen.add(path);
      return article;
    }
    var controller = new AbortController();
    async function fetchChunk(start, count){
      var feed = new URL(feedPath, url.origin);
      feed.searchParams.set('alt', 'json');
      feed.searchParams.set('orderby', 'published');
      feed.searchParams.set('start-index', String(start));
      feed.searchParams.set('max-results', String(count));
      var resp = await fetch(feed.href, {credentials:'same-origin',signal:controller.signal});
      if(!resp.ok){ throw new Error('Blogger feed unavailable'); }
      var json = await resp.json(), data = json.feed;
      if(!data){ throw new Error('Invalid Blogger feed'); }
      var countTotal = Number(data.openSearch$totalResults && data.openSearch$totalResults.$t);
      if(!Number.isSafeInteger(countTotal) || countTotal < 0){ throw new Error('Invalid feed total'); }
      return {total:countTotal, entries:Array.isArray(data.entry) ? data.entry : []};
    }
    async function loadBatch(start, length){
      var items = [], totalCount = null;
      // Some Blogger feeds return fewer items than requested; retry with offsets.
      for(var attempt = 0; attempt < length && items.length < length; attempt++){
        var batch = await fetchChunk(start + items.length, length - items.length);
        if(totalCount === null){ totalCount = batch.total; }
        if(!batch.entries.length || start + items.length > totalCount){ break; }
        var previous = items.length;
        items.push.apply(items, batch.entries);
        if(items.length === previous){ break; }
      }
      return {total:totalCount || 0, entries:items.slice(0,length)};
    }
    async function appendBatch(start, count){
      if(running){ return; } running = true;
      control.disabled = true; status.textContent = 'Đang tải bài viết…';
      try{
        var data = await loadBatch(start, count);
        total = data.total;
        var fragment = document.createDocumentFragment(), added = 0;
        data.entries.forEach(function(entry, index){
          var card = createCard(entry, index);
          if(card){ fragment.appendChild(card); added++; }
        });
        if(added){
          grid.appendChild(fragment); postThumbnails.observe(grid);
          grid.setAttribute('data-page-size', String(PAGE_SIZE));
        }
        nextIndex = start + data.entries.length;
        control.hidden = nextIndex > total || !data.entries.length;
        status.textContent = '';
        // Retain the native Older Posts link: it provides a server-rendered crawl path.
      }catch(e){
        status.textContent = 'Chưa thể tải thêm bài; có thể dùng liên kết trang cũ bên dưới.';
        control.hidden = false;
      }finally{ running = false; control.disabled = false; }
    }
    control.addEventListener('click', function(){ appendBatch(nextIndex, PAGE_SIZE); });
    appendBatch(1, PAGE_SIZE);
    window.addEventListener('pagehide', function(){ controller.abort(); }, {once:true});
  });

  // Catalog pagination uses feed offsets; native Blogger pages can be shortened automatically.
  onReady(function initCatalogPagination(){
    var locationUrl = new URL(window.location.href);
    // Keep Blogger's server-rendered first page intact. Replacing the grid while a reader
    // is clicking can cancel the first click and make article titles feel frozen.
    // Client-side catalog loading is retained only for explicit legacy cs-page URLs.
    if(!locationUrl.searchParams.has('cs-page')){ return; }
    var labelMatch = locationUrl.pathname.match(/^\/search\/label\/([^/]+)\/?$/);
    var homePath = locationUrl.pathname === '/' || locationUrl.pathname === '/search';
    if(!document.body.classList.contains('feed-view') || (!labelMatch && !homePath) ||
       locationUrl.searchParams.has('q') || !window.fetch || !window.AbortController){ return; }

    var widget = document.querySelector('.widget.Blog');
    if(!widget){ return; }
    var grid = widget.querySelector('.blog-posts');
    var pager = widget.querySelector('#blog-pager');
    if(!grid || !pager){ return; }
    var label;
    try{ label = labelMatch ? decodeURIComponent(labelMatch[1]) : ''; }catch(error){ return; }
    // Keep a page's offsets stable on resize; CSS reflows its existing cards.
    var PAGE_SIZE = 12;
    grid.setAttribute('data-page-size', String(PAGE_SIZE));
    var basePath = labelMatch ? '/search/label/' + encodeURIComponent(label) : '/';
    var feedPath = '/feeds/posts/default' + (label ? '/-/' + encodeURIComponent(label) : '');
    var notice = document.createElement('div');
    notice.className = 'catalog-notice';
    notice.setAttribute('role', 'status');
    grid.before(notice);
    grid.id = 'cs-post-catalog';
    grid.setAttribute('tabindex', '-1');
    var requestId = 0;
    var activeController;
    var displayedPage = null;
    var displayedSnapshot = null;
    var snapshotKey = 'cs-note-catalog-snapshot:' + basePath;

    function element(tag, className, value){
      var node = document.createElement(tag);
      if(className){ node.className = className; }
      if(value !== undefined){ node.textContent = value; }
      return node;
    }
    function readPage(url){
      var raw = url.searchParams.get('cs-page') || '1';
      var page = Number(raw);
      if(!/^\d+$/.test(raw) || !Number.isSafeInteger(page) || page < 1){ return 1; }
      var previousSize = Number(url.searchParams.get('cs-size')) === 9 ? 9 : 12;
      return Math.floor(((page - 1) * previousSize) / PAGE_SIZE) + 1;
    }
    function readSnapshot(url, navigation){
      var value = url.searchParams.get('cs-before');
      if(!value && navigation === 'push'){ value = displayedSnapshot; }
      if(!value && (navigation === 'pop' || url.searchParams.has('cs-page'))){
        var state = history.state;
        if(state && state.csBasePath === basePath){ value = state.csSnapshot; }
        if(!value){
          try{
            var saved = JSON.parse(sessionStorage.getItem(snapshotKey) || 'null');
            if(saved && Date.now() - saved.savedAt >= 0 && Date.now() - saved.savedAt < 3600000){ value = saved.snapshot; }
          }catch(error){ /* A fresh snapshot is safe when session storage is unavailable. */ }
        }
      }
      var date = value && new Date(value);
      return date && Number.isFinite(date.getTime()) && date.getTime() <= Date.now()
        ? date.toISOString() : new Date().toISOString();
    }
    function pageUrl(page){
      var url = new URL(basePath, locationUrl.origin);
      if(locationUrl.searchParams.has('m')){ url.searchParams.set('m', locationUrl.searchParams.get('m')); }
      if(page > 1){
        url.searchParams.set('max-results', String(PAGE_SIZE));
        url.searchParams.set('cs-page', String(page));
        url.searchParams.set('cs-size', String(PAGE_SIZE));
      }
      url.hash = 'cs-post-catalog';
      return url;
    }
    function safeUrl(value){
      try{
        var url = new URL(value, locationUrl.origin);
        return /^https?:$/.test(url.protocol) ? url.href : '';
      }catch(error){ return ''; }
    }
    function plainText(value){
      var template = document.createElement('template');
      template.innerHTML = value || '';
      template.content.querySelectorAll('script, style, iframe, object, svg, noscript').forEach(function(node){ node.remove(); });
      return template.content.textContent.replace(/\s+/g, ' ').trim();
    }
    function card(entry, index){
      var alternate = (entry.link || []).find(function(link){ return link.rel === 'alternate' && link.type === 'text/html'; });
      var href = alternate && safeUrl(alternate.href);
      if(!href){ throw new Error('Missing article URL'); }
      var title = plainText(entry.title && entry.title.$t) || 'Bài viết';
      var article = element('article', 'post-outer-container cs-catalog-card');
      var outer = element('div', 'post-outer');
      var thumbnail = postThumbnails.preview(entry);
      var cover = element('a', 'snippet-thumbnail' + (thumbnail ? '' : ' thumbnail-empty'));
      cover.href = href;
      cover.setAttribute('aria-label', title);
      cover.setAttribute('data-post-id', (entry.id && entry.id.$t || '').split('.post-').pop());
      if(thumbnail){
        postThumbnails.attach(cover, thumbnail, index === 0 ? 'high' : index < 4 ? 'visible' : false, function(){
          return postThumbnails.preview(entry, true);
        });
      }
      var content = element('div', 'post-content');
      var titleContainer = element('div', 'post-title-container');
      var heading = element('h3', 'post-title entry-title');
      var link = element('a', '', title);
      link.href = href;
      heading.appendChild(link);
      titleContainer.appendChild(heading);
      content.appendChild(titleContainer);
      var published = entry.published && new Date(entry.published.$t);
      if(published && Number.isFinite(published.getTime())){
        var metadata = element('div', 'post-header-container');
        var time = element('time', '', published.toLocaleDateString('vi-VN', {day:'2-digit', month:'2-digit', year:'numeric'}));
        time.dateTime = published.toISOString();
        metadata.appendChild(time);
        content.appendChild(metadata);
      }
      var excerpt = plainText((entry.summary || entry.content || {}).$t);
      if(excerpt){ content.appendChild(element('p', 'post-body', excerpt.length > 240 ? excerpt.slice(0, 237) + '...' : excerpt)); }
      outer.append(cover, content);
      article.appendChild(outer);
      return article;
    }
    async function readFeed(start, count, snapshot, signal){
      var url = new URL(feedPath, locationUrl.origin);
      url.searchParams.set('alt', 'json');
      url.searchParams.set('orderby', 'published');
      url.searchParams.set('start-index', String(start));
      url.searchParams.set('max-results', String(count));
      url.searchParams.set('published-max', snapshot);
      var response = await fetch(url.href, {credentials:'same-origin', signal:signal});
      if(!response.ok){ throw new Error('Feed unavailable'); }
      var data = await response.json();
      var feed = data.feed;
      var rawTotal = feed && feed.openSearch$totalResults && feed.openSearch$totalResults.$t;
      var total = Number(rawTotal);
      if(rawTotal === undefined || !Number.isSafeInteger(total) || total < 0 ||
         (feed.entry !== undefined && !Array.isArray(feed.entry))){ throw new Error('Invalid feed'); }
      var rawStart = feed.openSearch$startIndex && Number(feed.openSearch$startIndex.$t);
      if(rawStart && rawStart !== start && (feed.entry || []).length){ throw new Error('Feed offset ignored'); }
      return {total:total, entries:feed.entry || []};
    }
    async function readPageData(page, snapshot, signal){
      var start = (page - 1) * PAGE_SIZE + 1;
      if(!Number.isSafeInteger(start)){ page = 1; start = 1; }
      var batch = await readFeed(start, PAGE_SIZE, snapshot, signal);
      var total = batch.total;
      var pages = Math.max(1, Math.ceil(total / PAGE_SIZE));
      if(page > pages){ return readPageData(pages, snapshot, signal); }
      var entries = batch.entries.slice(0, PAGE_SIZE);
      var needed = Math.min(PAGE_SIZE, total - start + 1);
      // Fill partial feed responses without confusing a short response with the last page.
      while(entries.length < needed){
        batch = await readFeed(start + entries.length, needed - entries.length, snapshot, signal);
        if(!batch.entries.length || batch.total !== total){ throw new Error('Incomplete feed'); }
        entries = entries.concat(batch.entries).slice(0, needed);
      }
      var ids = new Set();
      entries.forEach(function(entry){
        var id = entry.id && entry.id.$t;
        if(!id || ids.has(id)){ throw new Error('Duplicate feed entry'); }
        ids.add(id);
      });
      return {page:page, pages:pages, total:total, entries:entries};
    }
    function pagination(data){
      var fragment = document.createDocumentFragment();
      var first = data.total ? (data.page - 1) * PAGE_SIZE + 1 : 0;
      var last = first ? first + data.entries.length - 1 : 0;
      fragment.appendChild(element('p', 'cs-page-summary', 'Trang ' + data.page + ' / ' + data.pages + ' · ' + first + '–' + last + ' / ' + data.total + ' bài viết'));
      function control(text, page, className, disabled, relation){
        var node = element(disabled ? 'button' : 'a', className, text);
        node.setAttribute('aria-label', className === 'cs-page-number' ? 'Trang ' + page : text);
        if(disabled){ node.type = 'button'; node.disabled = true; }
        else{
          node.href = pageUrl(page).href;
          node.dataset.csPage = String(page);
          node.setAttribute('aria-controls', grid.id);
          if(relation){ node.rel = relation; }
          if(page === data.page){ node.setAttribute('aria-current', 'page'); }
        }
        fragment.appendChild(node);
      }
      control('Trang trước', data.page - 1, 'blog-pager-newer-link', data.page === 1, 'prev');
      var previous = 0;
      var numbers = Array.from(new Set([1, data.page - 1, data.page, data.page + 1, data.pages]))
        .filter(function(number){ return number >= 1 && number <= data.pages; })
        .sort(function(a, b){ return a - b; });
      numbers.forEach(function(number){
        if(previous && number - previous > 1){ fragment.appendChild(element('span', 'cs-page-gap', '…')); }
        control(String(number), number, 'cs-page-number', false);
        previous = number;
      });
      control('Trang sau', data.page + 1, 'blog-pager-older-link', data.page === data.pages, 'next');
      return fragment;
    }
    async function loadPage(url, navigation){
      var token = ++requestId;
      if(activeController){ activeController.abort(); }
      var controller = new AbortController();
      activeController = controller;
      var timeout = setTimeout(function(){ controller.abort(); }, 15000);
      var page = readPage(url);
      var snapshot = readSnapshot(url, navigation);
      grid.setAttribute('aria-busy', 'true');
      notice.textContent = 'Đang tải bài viết…';
      try{
        var data = await readPageData(page, snapshot, controller.signal);
        if(token !== requestId){ return; }
        var cards = document.createDocumentFragment();
        for(var index = 0; index < data.entries.length; index++){
          cards.appendChild(card(data.entries[index], index));
          // Yield between rows so parsing technical articles does not block scrolling/input.
          if(index % 4 === 3){ await new Promise(function(resolve){ setTimeout(resolve, 0); }); }
          if(token !== requestId){ return; }
        }
        var controls = pagination(data);
        postThumbnails.release(grid);
        grid.replaceChildren(cards);
        grid.classList.remove('cs-catalog-hidden');
        widget.querySelectorAll('.no-posts, .no-posts-message, .status-msg-wrap, .post-filter-message').forEach(function(node){ node.classList.add('cs-native-empty-hidden'); });
        pager.replaceChildren(controls);
        pager.classList.add('cs-pagination');
        pager.setAttribute('aria-label', 'Phân trang bài viết');
        var nextUrl = pageUrl(data.page);
        if(navigation === 'initial'){
          nextUrl.hash = url.hash;
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	    var list = rail.querySelector('.related-posts-list');
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	    if(!window.fetch){
	      list.innerHTML = '
Related notes are unavailable in this browser.
';
	      return;
	    }

	    var labelNodes = Array.prototype.slice.call(rail.querySelectorAll('[data-related-label]'));
	    var labels = labelNodes.map(function(node){
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	    if(!labels.length){
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	    function cleanUrl(url){
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	    var requests = labels.map(function(label){
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';
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	        var link = document.createElement('a');
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    refreshAvatars();
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    var header = document.querySelector('.centered-top-container');
    var lastY = window.pageYOffset || document.documentElement.scrollTop || 0;
    var mobileHeaderQuery = window.matchMedia ? window.matchMedia('(max-width: 760px)') : null;
    var accumulatedUp = 0;
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      var isMobile = isMobileHeaderMode();
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      lastY = currentY;
      ticking = false;
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        ticking = true;
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})();

// Monetag vignette requests are counted across distinct, engaged article reads (default: two).
//]]>