Problem 46034. Construct the Seidel-Entringer-Arnold triangle

Created by ChrisR

Solve Later

{"parts":[{"partUri":"/matlab/document.xml","contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?><w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Several problems in Cody ask us to construct part or all of triangles in which entries follow a pattern. Cody Problems</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/37\"><w:r><w:t>37</w:t></w:r></w:hyperlink><w:r><w:t>,</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/1463\"><w:r><w:t>1463</w:t></w:r></w:hyperlink><w:r><w:t>,</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/44037\"><w:r><w:t>44037</w:t></w:r></w:hyperlink><w:r><w:t>, and</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/44904\"><w:r><w:t>44904</w:t></w:r></w:hyperlink><w:r><w:t> involve Pascal's triangle, which consists of the binomial coefficients, and Cody Problem</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/groups/31/problems/1845\"><w:r><w:t>1845</w:t></w:r></w:hyperlink><w:r><w:t> extends Pascal's triangle to a pyramid. Cody Problem</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/45423-bernoulli-s-triangle\"><w:r><w:t>45460</w:t></w:r></w:hyperlink><w:r><w:t> involves the Bernoulli triangle, which consists of partial sums of the binomial coefficients.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>This problem deals with the Seidel-Entringer-Arnold triangle (also called the Euler-Bernoulli triangle and the secant-tangent triangle). The first eight layers are</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[                         1\n                      0      1\n                  1      1      0\n               0      1      2      2\n           5      5      4      2      0\n        0      5     10     14     16     16\n    61    61     56     46     32     16      0\n  0    61    122    178    224    256    272    272]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>The name \"secant-tangent triangle\" arises because the sides contain the coefficients in the Taylor series for </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"sec(x)\"/></w:customXmlPr><w:r><w:t>\\sec(x)</w:t></w:r></w:customXml><w:r><w:t> and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"tan(x)\"/></w:customXmlPr><w:r><w:t>\\tan(x)</w:t></w:r></w:customXml><w:r><w:t>:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>       </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/><w:attr w:name=\"altTextString\" w:val=\"sec(x) = 1 + 1x^2/2! + 5x^4/4! + 61x^6/6! + ...\"/></w:customXmlPr><w:r><w:t>\\sec(x) = 1 + 1\\frac{x^2}{2!}+5\\frac{x^4}{4!}+61\\frac{x^6}{6!}+\\ldots</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>       </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/><w:attr w:name=\"altTextString\" w:val=\"tan(x) = 1x + 2x^3/3! + 16x^5/5! + 272x^7/7! + ...\"/></w:customXmlPr><w:r><w:t>\\tan(x) = 1x + 2\\frac{x^3}{3!}+16\\frac{x^5}{5!}+272\\frac{x^7}{7!}+\\ldots</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Construct the nth layer of this triangle.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Hint: Use the boustrophedon (or ox-plowing) rule.</w:t></w:r></w:p></w:body></w:document>","relationship":null}],"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","target":"/matlab/document.xml","relationshipId":"rId1"}]}

<div style = "text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; "><div style="block-size: 510.567px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 255.283px; transform-origin: 407px 255.283px; vertical-align: baseline; "><div style="block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 359.033px 7.8px; transform-origin: 359.033px 7.8px; unicode-bidi: normal; "><span style="">Several problems in Cody ask us to construct part or all of triangles in which entries follow a pattern. Cody Problems</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "><span style=""> </span></span><a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/37"><span style=""><span style="">37</span></span></a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "><span style="">,</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "><span style=""> </span></span><a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/1463"><span style=""><span style="perspective-origin: 15.5667px 7.8px; transform-origin: 15.5667px 7.8px; ">1463</span></span></a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "><span style="">,</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "><span style=""> </span></span><a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/44037"><span style=""><span style="">44037</span></span></a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5667px 7.8px; transform-origin: 15.5667px 7.8px; unicode-bidi: normal; "><span style="">, and</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "><span style=""> </span></span><a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/44904"><span style=""><span style="">44904</span></span></a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 271.55px 7.8px; transform-origin: 271.55px 7.8px; unicode-bidi: normal; "><span style=""> involve Pascal's triangle, which consists of the binomial coefficients, and Cody Problem</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "><span style=""> </span></span><a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/groups/31/problems/1845"><span style=""><span style="">1845</span></span></a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "><span style=""> extends Pascal's triangle to a pyramid. Cody Problem</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "><span style=""> </span></span><a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/45423-bernoulli-s-triangle"><span style=""><span style="">45460</span></span></a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 191.383px 7.8px; transform-origin: 191.383px 7.8px; unicode-bidi: normal; "><span style=""> involves the Bernoulli triangle, which consists of partial sums of the binomial coefficients.</span></span></div><div style="block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 382.033px 7.8px; transform-origin: 382.033px 7.8px; unicode-bidi: normal; "><span style="">This problem deals with the Seidel-Entringer-Arnold triangle (also called the Euler-Bernoulli triangle and the secant-tangent triangle). The first eight layers are</span></span></div><div style="background-color: rgb(247, 247, 247); block-size: 163.467px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 81.7333px; transform-origin: 404px 81.7333px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; "><div style="background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.916667px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.916667px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.916667px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.916667px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 100.1px 8.25px; transform-origin: 100.1px 8.25px; unicode-bidi: normal; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">                         1</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.916667px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.916667px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.916667px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.916667px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 115.5px 8.25px; transform-origin: 115.5px 8.25px; unicode-bidi: normal; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">                      0      1</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.916667px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.916667px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.916667px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.916667px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 127.05px 8.25px; transform-origin: 127.05px 8.25px; unicode-bidi: normal; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">                  1      1      0</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.916667px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.916667px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.916667px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.916667px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 142.45px 8.25px; transform-origin: 142.45px 8.25px; unicode-bidi: normal; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">               0      1      2      2</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.916667px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.916667px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.916667px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.916667px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 154px 8.25px; transform-origin: 154px 8.25px; unicode-bidi: normal; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">           5      5      4      2      0</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.916667px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.916667px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.916667px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.916667px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 169.4px 8.25px; transform-origin: 169.4px 8.25px; unicode-bidi: normal; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">        0      5     10     14     16     16</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.916667px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.916667px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.916667px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.916667px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 180.95px 8.25px; transform-origin: 180.95px 8.25px; unicode-bidi: normal; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">    61    61     56     46     32     16      0</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.916667px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.916667px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.916667px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.916667px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 196.35px 8.25px; transform-origin: 196.35px 8.25px; unicode-bidi: normal; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">  0    61    122    178    224    256    272    272</span></span></div></div><div style="block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 333.917px 7.8px; transform-origin: 333.917px 7.8px; unicode-bidi: normal; "><span style="">The name "secant-tangent triangle" arises because the sides contain the coefficients in the Taylor series for </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" alt="sec(x)" style="width: 42px; height: 18.5px;" width="42" height="18.5"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5667px 7.8px; transform-origin: 15.5667px 7.8px; unicode-bidi: normal; "><span style=""> and </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" alt="tan(x)" style="width: 41.5px; height: 18.5px;" width="41.5" height="18.5"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "><span style="">:</span></span></div><div style="block-size: 36.05px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.0333px; text-align: left; transform-origin: 384px 18.0333px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6px 7.8px; transform-origin: 13.6px 7.8px; unicode-bidi: normal; "><span style="">       </span></span><span style="vertical-align:-15px"><img src="data:image/png;base64,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" alt="sec(x) = 1 + 1x^2/2! + 5x^4/4! + 61x^6/6! + ..." style="width: 226px; height: 36px;" width="226" height="36"></span></div><div style="block-size: 36.05px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.0333px; text-align: left; transform-origin: 384px 18.0333px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6px 7.8px; transform-origin: 13.6px 7.8px; unicode-bidi: normal; "><span style="">       </span></span><span style="vertical-align:-15px"><img src="data:image/png;base64,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" alt="tan(x) = 1x + 2x^3/3! + 16x^5/5! + 272x^7/7! + ..." style="width: 247.5px; height: 36px;" width="247.5" height="36"></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 117.85px 7.8px; transform-origin: 117.85px 7.8px; unicode-bidi: normal; "><span style="">Construct the nth layer of this triangle.</span></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 152.883px 7.8px; transform-origin: 152.883px 7.8px; unicode-bidi: normal; "><span style="">Hint: Use the boustrophedon (or ox-plowing) rule.</span></span></div></div></div>

Several problems in Cody ask us to construct part or all of triangles in which entries follow a pattern. Cody Problems <https://www.mathworks.com/matlabcentral/cody/problems/37 37>, <https://www.mathworks.com/matlabcentral/cody/problems/1463 1463>, <https://www.mathworks.com/matlabcentral/cody/problems/44037 44037>, and <https://www.mathworks.com/matlabcentral/cody/problems/44904 44904> involve Pascal's triangle, which consists of the binomial coefficients, and Cody Problem <https://www.mathworks.com/matlabcentral/cody/groups/31/problems/1845 1845> extends Pascal's triangle to a pyramid. Cody Problem <https://www.mathworks.com/matlabcentral/cody/problems/45423-bernoulli-s-triangle 45460> involves the Bernoulli triangle, which consists of partial sums of the binomial coefficients.

This problem deals with the Seidel-Entringer-Arnold triangle (also called the Euler-Bernoulli triangle and the secant-tangent triangle). The first eight layers are

1
                      0      1
                  1      1      0
               0      1      2      2
           5      5      4      2      0
        0      5     10     14     16     16
    61    61     56     46     32     16      0
  0    61    122    178    224    256    272    272   
  
The name "secant-tangent triangle" arises because the sides contain the coefficients in the Taylor series for sec(x) and tan(x):

sec(x) = 1 + 1x^2/2! + 5x^4/4! + 61x^6/6! + ...
 tan(x) = 1x + 2x^3/3! + 16x^5/5! + 272x^7/7! + ...

Construct the nth layer of this triangle.

Hint: Use the boustrophedon (or ox-plowing) rule.

Solve

Solution Stats

30 Solutions
17 Solvers

Last Solution submitted on Feb 05, 2023

Last 200 Solutions

Problem Comments

Problem Recent Solvers17

Problem Tags

Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!

Problem 46034. Construct the Seidel-Entringer-Arnold triangle

Solution Stats

Last 200 Solutions

Problem Comments

Problem Recent Solvers17

Suggested Problems

More from this Author192

Problem Tags

Community Treasure Hunt

Problem 46034. Construct the Seidel-Entringer-Arnold triangle

Solution Stats

Last 200 Solutions

Problem Comments

Problem Recent Solvers17

Suggested Problems

More from this Author192

Problem Tags

Community Treasure Hunt

Players