Double sum over $leftlfloor{ac+bdover k}rightrfloor$












1














We have
$$leftlfloor{ac+bdover k}rightrfloor-leftlfloor{ac+bd-1over k}rightrfloor=1-leftlceil{ (ac+bd)mod{k}over k}rightrceil$$
for $a,b,c,d,k$ - integers, $ageqslant0$, $bgeqslant0$, $c>0$, $d>0$, $k>0$. Next we want to calculate
$$sumlimits_{a=0}^{n}sumlimits_{b=0}^{m}leftlfloor{ac+bdover k}rightrfloor$$
or
$$sumlimits_{a=0}^{n}sumlimits_{b=0}^{m}leftlceil{(ac+bd)mod{k}over k}rightrceil$$
Is there a nice closed form for it?






summation closed-form floor-function ceiling-function







share|cite|improve this question





























    1














    We have
    $$leftlfloor{ac+bdover k}rightrfloor-leftlfloor{ac+bd-1over k}rightrfloor=1-leftlceil{ (ac+bd)mod{k}over k}rightrceil$$
    for $a,b,c,d,k$ - integers, $ageqslant0$, $bgeqslant0$, $c>0$, $d>0$, $k>0$. Next we want to calculate
    $$sumlimits_{a=0}^{n}sumlimits_{b=0}^{m}leftlfloor{ac+bdover k}rightrfloor$$
    or
    $$sumlimits_{a=0}^{n}sumlimits_{b=0}^{m}leftlceil{(ac+bd)mod{k}over k}rightrceil$$
    Is there a nice closed form for it?






    summation closed-form floor-function ceiling-function







    share|cite|improve this question



























      1












      1








      1







      We have
      $$leftlfloor{ac+bdover k}rightrfloor-leftlfloor{ac+bd-1over k}rightrfloor=1-leftlceil{ (ac+bd)mod{k}over k}rightrceil$$
      for $a,b,c,d,k$ - integers, $ageqslant0$, $bgeqslant0$, $c>0$, $d>0$, $k>0$. Next we want to calculate
      $$sumlimits_{a=0}^{n}sumlimits_{b=0}^{m}leftlfloor{ac+bdover k}rightrfloor$$
      or
      $$sumlimits_{a=0}^{n}sumlimits_{b=0}^{m}leftlceil{(ac+bd)mod{k}over k}rightrceil$$
      Is there a nice closed form for it?






      summation closed-form floor-function ceiling-function







      share|cite|improve this question















      We have
      $$leftlfloor{ac+bdover k}rightrfloor-leftlfloor{ac+bd-1over k}rightrfloor=1-leftlceil{ (ac+bd)mod{k}over k}rightrceil$$
      for $a,b,c,d,k$ - integers, $ageqslant0$, $bgeqslant0$, $c>0$, $d>0$, $k>0$. Next we want to calculate
      $$sumlimits_{a=0}^{n}sumlimits_{b=0}^{m}leftlfloor{ac+bdover k}rightrfloor$$
      or
      $$sumlimits_{a=0}^{n}sumlimits_{b=0}^{m}leftlceil{(ac+bd)mod{k}over k}rightrceil$$
      Is there a nice closed form for it?






      summation closed-form floor-function ceiling-function




      summation closed-form floor-function ceiling-function






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited 2 days ago









      Key Flex

      7,52441232




      7,52441232










      asked 2 days ago









      user514787

      695210




      695210






















          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
          });


          }
          });














          draft saved

          draft discarded


















          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3060721%2fdouble-sum-over-left-lflooracbd-over-k-right-rfloor%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
















          draft saved

          draft discarded




















































          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.





          Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


          Please pay close attention to the following guidance:


          • 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.


          To learn more, see our tips on writing great answers.




          draft saved


          draft discarded














          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3060721%2fdouble-sum-over-left-lflooracbd-over-k-right-rfloor%23new-answer', 'question_page');
          }
          );

          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







          -

          Popular posts from this blog

          1300-talet

          Image
          Den här artikeln handlar om hundraårsperioden 1300–1399. För decenniet 1300–1309, se 1300-talet (decennium). .mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output ...
          Read more

          1300-talet

          Image
          Den här artikeln handlar om hundraårsperioden 1300–1399. För decenniet 1300–1309, se 1300-talet (decennium). .mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output ...
          Read more

          Display a custom attribute below product name in the front-end Magento 1.9.3.8

          Image
          0 I need to display a custom attribute bellow the product name and I don't know how can I call this attribute. I am editing the file /app/design/frontend/templatemela/MAG100219_1/template/catalog/product/view.phtml But how can I call this attribute in php? I tried calling a string echo with the name I put in the attribute, but it didn't work. Code is as below: <div class="page-title product-title"> <h1><?php echo $_helper->productAttribute($_product, $_product->getName(), 'name') ?></h1> </div> <div class="codigoinmetro"> <h2><?php echo $_helper->productAttribute($_product, $_product->getName(), 'registroinmetro') ?></h2> </div> php attrib...
          Read more