Problem 57555. Easy Sequences 91: Generalized McCarthy-91 Recursive Function

Created by Ramon Villamangca

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>The </w:t></w:r><w:hyperlink w:docLocation=\"https://en.wikipedia.org/wiki/McCarthy_91_function\"><w:r><w:t>McCarthy 91 function</w:t></w:r></w:hyperlink><w:r><w:t> is a recursive function, defined by the computer scientist John McCarthy as a test case for formal verification within computer science.</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:rPr><w:b/></w:rPr><w:t>The McCarthy 91 function is defined for integer </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t> as:</w:t></w:r><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:customXmlPr><w:r><w:t>M(n) = n-10 \\ \\text{if} \\ n &gt; 100 \\ \\text{and} \\ \nM(M(n+11)) \\ \\text{if} \\ n \\le 100</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>Remarkably, the function yields </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>M(n) = 91</w:t></w:r></w:customXml><w:r><w:t> for all </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>n \\le 101</w:t></w:r></w:customXml><w:r><w:t> (hence, the function name).</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:rPr><w:b/></w:rPr><w:t>A more generalized form of the McCarthy's recursive function is defined below: </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:customXmlPr><w:r><w:t>G(n) = n-b \\ \\text{if} \\ n &gt; a \\ \\text{and} \\ \nG^k(n+c) \\ \\text{if} \\ n \\le a</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>For positive integers </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>a</w:t></w:r></w:customXml><w:r><w:t>, </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>b</w:t></w:r></w:customXml><w:r><w:t>, </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>c</w:t></w:r></w:customXml><w:r><w:t>, </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>k</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:customXmlPr><w:r><w:t>n</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>The expression </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>G^k(n+c) </w:t></w:r></w:customXml><w:r><w:t>, means applying the function </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>G</w:t></w:r></w:customXml><w:r><w:t>, </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>k</w:t></w:r></w:customXml><w:r><w:t> number of times:</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:customXmlPr><w:r><w:t>G^k(x) = \\underbrace{G(G(G(...G(x))))}\\\\_{\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\text{k times}}</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>This means that the 'original' McCarthy-91 function </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>M(n)</w:t></w:r></w:customXml><w:r><w:t> is actually </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>G(n)</w:t></w:r></w:customXml><w:r><w:t>, with </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>a=100</w:t></w:r></w:customXml><w:r><w:t>, </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>b=10</w:t></w:r></w:customXml><w:r><w:t>, </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>c=11</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:customXmlPr><w:r><w:t>k=2</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:rPr><w:b/></w:rPr><w:t>Given integers, </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>a</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t>, </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>b</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t>, </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>c</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t>, </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>k</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t>, and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t>, evaluate the value of </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>G(n)</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><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:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>NOTE:</w:t></w:r><w:r><w:t> The McCarthy-91 function was created to test recursive function implementations. In this problem, while iterative loops are allowed, use of recursion is encouraged.</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: 454px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 227px; transform-origin: 407px 227px; vertical-align: baseline; "><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: 14.5px 8px; transform-origin: 14.5px 8px; unicode-bidi: normal; ">The <a target='_blank' href = "https://en.wikipedia.org/wiki/McCarthy_91_function">McCarthy 91 function</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: 286.5px 8px; transform-origin: 286.5px 8px; unicode-bidi: normal; "> is a recursive function, defined by the computer scientist John McCarthy as a test case for formal verification within computer science.</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: 167.5px 8px; transform-origin: 167.5px 8px; unicode-bidi: normal; ">The McCarthy 91 function is defined for integer n<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: 12px 8px; transform-origin: 12px 8px; unicode-bidi: normal; "> as:<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: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; "> </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: 80px 8px; transform-origin: 80px 8px; unicode-bidi: normal; "> <img src="data:image/png;base64,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" style="width: 366.5px; height: 18.5px;" width="366.5" height="18.5"></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: 100px 8px; transform-origin: 100px 8px; unicode-bidi: normal; ">Remarkably, the function yields <img src="data:image/png;base64,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" style="width: 70.5px; height: 18.5px;" width="70.5" height="18.5"><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: 21.5px 8px; transform-origin: 21.5px 8px; unicode-bidi: normal; "> for all <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGcAAAAkCAYAAACUjSQ7AAAEKElEQVRoQ+1auaoVQRD1fYC4RmLkEgiCgkskgiaKhgYqJoLimop75i6YuQbCy9xiRRMNNBFFFAMDl9DI7Q/0nEcXFEMv1T1z7/TFaTjc+2a6u6rrVFVX931Ts4ZWrQWmqtVsUGzWQE7FTjCQM5BTsQUqVm2InIGcai2wAJrNAb5laMgxK1z/T/j8mTFWuprk/q+RQ+PsA04Cp4A7BgNvQJ+zAMc+cqQexuct4K6R4Cy5k0jONhjidaHHkoPjjpR5jpBDBnJ2oc894BmwR8lehe/vgd/A2gRB2XIniRwa6DywFFgNfDB4u+6yBH8wWj4CGwF6PVuKHCGAfX1yafQrwFtgnUenUrkTcc7RpHDtD4DTCS9N8cYU9dJIzlP025Iw/lfjXDlyqybHR8p1GOFVyvKG91Yj6ai5iHnPBOb+gueM6FD0yDCr3Jn+Naa1UZKSa6QLLko5bjdwP0AOn+9070hSqPrrnBxWGIsBneOZRxcB31umF1mrVDEHnAfyOdNXV5HStKnVSJLSOD62z8m+kyLRKjcYOayGVgKbgfUAqxrZNEnKTYA5mI1VyqYGcU1DxP7WpaVUT6MkRXSxGumXW38OObH0Z5UbJEeiYlp5MUN1NvACuOxWyAqFLaZMiJi+SMkl569agDVy6FxMzb7WmhyZ9A2+sHZnbX8E4GYnUaKFxHKxT8GDeHhJeSTnZ4ncxUYfcobmc6uRhBxmiPmRyfV8IyeHnv3DKcOUth+4BsiGSAPfdu8X4jPnCoOReQyQcwanYfRZT9lWAmL9SshZHlknt4LHTuDIydHC6NmsPhg90rjv0Lh8t7XQWr7UNi6ScsnhEqtJa2J8KsUD1rIGAbJRnsDzq4XkyLA+SLKSU1KtxW4crHJnbBM658ihin22A08UAanrjFKuQiQ9xIS5VzUpHaxG0k5qjZymvbQuVrlBcrTxfflT9pvUJpkyUOi978zD9Mko7Yokq5HkwpO6xiJCIix1AWqVGyRHb/Y+LxBFeFWu96FSMmLjmrcFXZFkNRId5TPAM1hsvZLmY8UA12mVGyRHe0GzfNRVnJTQrOBYGnfl1T6ymiSV3EqXphc5/TMqfBWbLp5SerUmR2p7n6foMGcJfQ6g0qELwa6jSUja0dIZsowEWXLm86U2cWZLcZQlt1kQaC/wpTSd8ngoZSUXOg13TYyej2elnJ+W9VhG/w1ALiqZio4CsbMax5AEHsolY/AZnZNHCsstSbbcJjk09F63Et/5RQTMRZ9pIHRLO0pi2szNFMU7w2b7gwfvAB6EYyTRPvyhjs7B9hzgPphK6UVy+/jJgNUg7+natNJ/rGgjc+xj+yBHH+xKF5zaeEvnrWpcH+TwB6w1La3AEr50z2kpenzD+yBnfKubcEkDORUTOJAzkFOxBSpWbYicgZyKLVCxav8AIqMlNEi4i8QAAAAASUVORK5CYII=" style="width: 51.5px; height: 18px;" width="51.5" height="18"><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: 89px 8px; transform-origin: 89px 8px; unicode-bidi: normal; "> (hence, the function name).</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: 280px 8px; transform-origin: 280px 8px; unicode-bidi: normal; ">A more generalized form of the McCarthy's recursive function is defined below: </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: 80px 8px; transform-origin: 80px 8px; unicode-bidi: normal; "> <img src="data:image/png;base64,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" style="width: 291px; height: 19.5px;" width="291" height="19.5"></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: 66.5px 8px; transform-origin: 66.5px 8px; unicode-bidi: normal; ">For positive integers 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: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; ">, b<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: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; ">, c<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: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; ">, k<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: 18px 8px; transform-origin: 18px 8px; unicode-bidi: normal; ">, and n<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: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; ">. </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: 50.5px 8px; transform-origin: 50.5px 8px; unicode-bidi: normal; ">The expression <img src="data:image/png;base64,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" style="width: 61px; height: 19.5px;" width="61" height="19.5"><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: 94.5px 8px; transform-origin: 94.5px 8px; unicode-bidi: normal; ">, means applying the function G<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: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; ">, k<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: 54.5px 8px; transform-origin: 54.5px 8px; unicode-bidi: normal; "> number of times:</div><div style="block-size: 52px; 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 26px; text-align: left; transform-origin: 384px 26px; 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: 80px 8px; transform-origin: 80px 8px; unicode-bidi: normal; "> <img src="data:image/png;base64,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" style="width: 165px; height: 52px;" width="165" height="52"></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: 162px 8px; transform-origin: 162px 8px; unicode-bidi: normal; ">This means that the 'original' McCarthy-91 function <img src="data:image/png;base64,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" style="width: 37px; height: 18.5px;" width="37" height="18.5"><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: 35px 8px; transform-origin: 35px 8px; unicode-bidi: normal; "> is actually <img src="data:image/png;base64,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" style="width: 33px; height: 18.5px;" width="33" height="18.5"><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: 18px 8px; transform-origin: 18px 8px; unicode-bidi: normal; ">, with <img src="data:image/png;base64,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" style="width: 51.5px; height: 18px;" width="51.5" height="18"><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: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; ">, <img src="data:image/png;base64,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" style="width: 44px; height: 18px;" width="44" height="18"><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: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; ">, <img src="data:image/png;base64,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" style="width: 43.5px; height: 18px;" width="43.5" height="18"><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: 16px 8px; transform-origin: 16px 8px; unicode-bidi: normal; "> and <img src="data:image/png;base64,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" style="width: 36px; height: 18px;" width="36" height="18"><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: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; ">.</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: 56.5px 8px; transform-origin: 56.5px 8px; unicode-bidi: normal; ">Given integers, 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: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; ">, b<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: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; ">, c<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: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; ">, k<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: 19px 8px; transform-origin: 19px 8px; unicode-bidi: normal; ">, and n<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: 80.5px 8px; transform-origin: 80.5px 8px; unicode-bidi: normal; ">, evaluate the value of <img src="data:image/png;base64,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" style="width: 33px; height: 18.5px;" width="33" height="18.5"><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: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; ">.</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: 50px 8px; transform-origin: 50px 8px; unicode-bidi: normal; ">--------------------</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: 21px 8px; transform-origin: 21px 8px; unicode-bidi: normal; ">NOTE:<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: 355px 8px; transform-origin: 355px 8px; unicode-bidi: normal; "> The McCarthy-91 function was created to test recursive function implementations. In this problem, while iterative loops are allowed, use of recursion is encouraged.</div></div></div>

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Last Solution submitted on Feb 19, 2023

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Solution 10285360

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1 Comment

Ramon Villamangca on 21 Jan 2023

This is an excellent solution. But recursive solution is encouraged.

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Problem 57555. Easy Sequences 91: Generalized McCarthy-91 Recursive Function

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Problem 57555. Easy Sequences 91: Generalized McCarthy-91 Recursive Function

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