Problem 47073. Find the nth Fibbinary number

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>The numbers 0, 1, 2, 4, 5, 8, 9, and 10 form the first eight elements (i.e., elements </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=\"a_0\"/></w:customXmlPr><w:r><w:t>a_0</w:t></w:r></w:customXml><w:r><w:t> to </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=\"a_7\"/></w:customXmlPr><w:r><w:t>a_7</w:t></w:r></w:customXml><w:r><w:t>) of the Fibbinary sequence. The name is a portmanteau that arose because of connections to Fibonacci numbers and binary numbers. The connection to Fibonacci numbers </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=\"F(k)\"/></w:customXmlPr><w:r><w:t>F(k)</w:t></w:r></w:customXml><w:r><w:t>is that if the </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/47058-determine-the-zeckendorf-expansion-of-a-number\"><w:r><w:t>Zeckendorf expansion</w:t></w:r></w:hyperlink><w:r><w:t> of </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=\"n\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:t> is </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=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"n = F(k_1) + F(k_2) + \\ldots + F(k_m)\"/></w:customXmlPr><w:r><w:t>n = F(k_1) + F(k_2) + \\ldots + F(k_m)</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>then</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=\"a_n = 2^{k_1-2}+2^{k_2-2}+\\ldots+2^{k_m-2}\"/></w:customXmlPr><w:r><w:t>a_n = 2^{k_1-2}+2^{k_2-2}+\\ldots+2^{k_m-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:t>The connection to binary numbers is that the binary representations of the Fibbinary numbers have no adjacent 1s. </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>For example, </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_7</w:t></w:r></w:customXml><w:r><w:t> = 10. The Zeckendorf expansion of 7 is 2+5, or </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=\"F(3)+F(5)\"/></w:customXmlPr><w:r><w:t>F(3)+F(5)</w:t></w:r></w:customXml><w:r><w:t>, and </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=\"10=2^{3-2}+2^{5-2} = 2+8\"/></w:customXmlPr><w:r><w:t>10=2^{3-2}+2^{5-2} = 2+8</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>Also, the binary expansion of 10 is 1010, which has no adjacent 1s. </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>Write a function to find the </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>th Fibbinary number. </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: 308.75px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 154.375px; transform-origin: 407px 154.375px; vertical-align: baseline; "><div style="block-size: 63.25px; 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 31.625px; text-align: left; transform-origin: 384px 31.625px; 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: 255.117px 7.91667px; transform-origin: 255.117px 7.91667px; unicode-bidi: normal; "><span style="">The numbers 0, 1, 2, 4, 5, 8, 9, and 10 form the first eight elements (i.e., elements </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAB8AAAAoCAYAAAAG0SEsAAABOUlEQVRYhe2VXbGDMBBGj4c4wEANREEV1AEO4qAW0ICEeLgWqgEL3Idkh21uEmiB2+lMzkweYMl+2T8CjUaj0Wh8iAvQAw6w8Z2Nz5czRT3wiOIWGIAfYI6rO0P4Fp0/AJPYZmX7V2Etfj9a+Kqc24y9W7G/jQGm6HgsfCNZmfibFUPoCVmeFxrSsURV2uSj3RcOPqpD9Su+npCoU8eCVYdziW0gX4qJMB1Vao4FTz4zRr1PSyF7qv2hU37N2HuWzKT1loPnIhS/1cmoid8IYzfyXJYh7pO9NfFSKYGQxll9aOJyLPMukUs3y0RsEV+tu0Sml+7eKXm/RWCzOISUO0KN0/926TKRkdot/g67G24vUo4UGbXcBB1G7SdTuqAOQ6ZBj5T0wqlRCx3LhXInTMShN1+j0fgefgFsKIIPBf1eagAAAABJRU5ErkJggg==" alt="a_0" style="width: 15.5px; height: 20px;" width="15.5" height="20"></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: 9.71667px 7.91667px; transform-origin: 9.71667px 7.91667px; unicode-bidi: normal; "><span style=""> to </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAB8AAAAoCAYAAAAG0SEsAAABJ0lEQVRYhe2WUY3DMAyGPw5lMAIlEARFMAZhUAajEAyFEA6lUAylsHuYc3Ozutmm5aaT8kl+aGz5d+y0KTQajUaj8SV6wAMj4GTNyXNfUzQCi4g7IAAzcBU71RA+S/IF6DLfVfn+VFiLXz4tPKjkbsd/KvjfpgNWSTwZMakrK9uuBFXUnhULHVWwdYqj+GNW9CKFucxm7PFtSLuOht+p4ka17kXYyhlKwlZiTaTcGU06P0MpULd8L9hz70w+b4vwbOyR+Jnb3Ca2YwnYXUKErYO7oVfiUartJHk6MGnnQewoccpnnYUH0s60TdzbtmbrR1wk7pnx/DJw263n8bv9ymWyYL81VUlfQf8NcU/FG6/ELPbSvD9BavnRK1iNdPFU+8NpNBr/gx+twmvaJTM4bAAAAABJRU5ErkJggg==" alt="a_7" style="width: 15.5px; height: 20px;" width="15.5" height="20"></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: 102.683px 7.91667px; transform-origin: 102.683px 7.91667px; unicode-bidi: normal; "><span style="">) of the Fibbinary sequence. The name is a portmanteau that arose because of connections to Fibonacci numbers and binary numbers. The connection to Fibonacci numbers </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" alt="F(k)" style="width: 32.5px; height: 19px;" width="32.5" height="19"></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: 37.7167px 7.91667px; transform-origin: 37.7167px 7.91667px; unicode-bidi: normal; "><span style="">is that if the </span></span><a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/47058-determine-the-zeckendorf-expansion-of-a-number"><span style=""><span style="">Zeckendorf expansion</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: 9.71667px 7.91667px; transform-origin: 9.71667px 7.91667px; unicode-bidi: normal; "><span style=""> of </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">n</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: 8.94167px 7.91667px; transform-origin: 8.94167px 7.91667px; unicode-bidi: normal; "><span style=""> is </span></span></div><div style="block-size: 21.25px; 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 10.625px; text-align: left; transform-origin: 384px 10.625px; 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: 9.70833px 7.91667px; transform-origin: 9.70833px 7.91667px; unicode-bidi: normal; "><span style="">     </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAY4AAAAoCAYAAADpNh07AAAJQklEQVR4nO2dbZHrOgyGXw5lsARKoAgWQRmUwTIohWIohHJYCsVQCvf+SN+J4nVsKbHSeI6emcw502bzYcmSLMsuEARBEARBEARBEARBEARBEARBEARBEARBsIArgOOG9/sBcNrwfr0Scgn2xgHA7f3vpwg9rfCFoZGWHBflPW4AzoX7n97XugL4fV97LQcA98J9984Jy+XyrbxHSS5H8Qw3AE+06Ui9y+Vf54zleqkJUA4AHoVzTxj0+weDHj3h42BCT5V8A/hPHE8MhjwV/kOcc1dc94ayIzi+v3+K67by9AcMjkhrSPfGAUPbSLnQscqDDpfnaBx6TS7snLzmC+06aO9y+df5wtD3pV7e8VcvGXDwnJrjoF7M9f8DBkN+E9d8rHiPGqGnSqQilDzt6X1ObWRwwdDwGqgMLQ0UMCj56/1vj7CteZTeg4a+5ngtcmHHb91Be5fLv44MKp6Vcx/Q9es7hiCoxpe4d4vsRO1eoacFLAYK73NKnpjC1Y4eGDFrRjFWOFLy4PQ+vPKxlg5KGZaexSKXg7i3Ni1poWe57JkDxvf3QmYebpVzf1APVM7QB40yO7LF/JynnnbPBdN0SI2agbpB39jeBorX98hXeiuwTAnUorET6rKzyMW7g/Yslz3DAOLleI8XxjaupXKYTi3xhH70cIX/+0k89bR7ZAShGS6WOiSjWm1ucIsI4gZ9esaC93PLUaCmPVvKZYsO2qtc9oy340izE7VRwgHlDMZZeR3imZ2Yw0tPu0cqQm6Iaxn60uBo2cJA0Tm1NiSeBkp20Llh/Bn6/KtVLlt00B7lsne8HYdMn86NXi1zD7+F66R4ZyfmWK2nx/dFZNTG2X5LKeSe0BgoSznmE7acICdgaaBkjlYea/LVVLjWk2meBqrWQTlxp20Xi1zkBORFfJaTyxp6lMve8XYcMjuRk9sZdj3Typ+jEylblo7Lo7XcF+npBdPGkvmub0zzfZZUALBuHYU81nRgRqJzBuoGvRJaFUEaKLYpHQcdyhVtJjmtDk2Dp4GSJbZpe7JUUDsaWNNBOaL5whh5vdAu4utNLnvH03HIiD+XnTjCphvUM639YvWlLBRhST8/9yoKWKyn7MiM8s4YH1Tm6TVzBCTNFy491nRiaaAeGCezHhiNd61yIn0fq+JIAwUMytBaCVgW2BIvA5V2UK674OQ2gxXthJ1VLrkOCoylvC2rlXqSSw94Oo50vRcD1zumwbU2fUqDr5XTnD26Zz5rzWI9lSkVRsTsQNIBWIxdqxHH0g6SGii56I8jDcs7adcSkLkIorVxAsYKpZZ4GSjZQV+YyvouPte2kVUuuQ56hs98R09y6QFPxyGzE+liVAaglqiczkZDLjsBDDq6xXzHIj1Nc75pNMyO2XoBmzcy4s8p2gW+BoqOiQbqG36RgzW6AepbLOQcbu6wIlfH5ow1IzwtFrkc8beD/sCvc/Ykl71QeifqThpwpMeS8lK5EjynD7ynFovjkEsGvjDuabXVnlJL9HRiYJ/422j0RluWiLWgZqB+Zj6fY42B+kF9kRuh0lhK5KxODfg7t7XkWBL51TroY+bzOSzvLjvoEbo0AOXB57YYkJ7kshfWvvsSWyWD5zkDajWsFsdBG/vEmJXYcm3FEj2dLMTKRXqlTr5npIHKCeEy8/kcSw2UTL+U8qMHjHl2Dpc9no1oI9tb4RyrTqQdNNce95nP57C8u+ygmnbmRP0V44aVHElqRkW9yGVPaEYc/1XOs1aApsFzDutaB4vjkDrF/1vmk9eyyHHwQV/461Hl/IZ1T5NPVlXJiH/Js+ewNK6MGmslfoSLidjmFkVlh2pJKfpainSo2hFYDYtcZEQuCyfm/pY7G0ukbtXaphe59ILXHIcMnmsjUC1axyFt7BPTke1WUwNmPZUPnWswThgt6eSfrKryMFCc1NVEM2mb0jlrtkpe4jgs0Y0WDwPl0UG1cpH6eMbUAcylNuZGP3Q6tXv2Ipde8HIcctlBqxQRbWetv8tR5BemtmurkaNZT2UlQU4R2UEYdVki909WVVn2QdJCQ1PLb0sDRcMilaOmmEsch2WFqhYPA2XZB0iLVi5SBuzMS8os5d/V2qYXufSCh+NIsxOtonw6gNpImLrE/n7ANNDcArOe0jHkHlDmo7m6ufRDJHvCw0DxurWJN7nNCJVQtqV0CCx/lixxHB450dYGyquDAjq55EoqZW6b7cd5jdLzMaVQe4ce5AJMCwVK79T6PCsejsO6CaoW6ntp1CCXDMjAR87l0H4dMeouC3t+MbaxrDgDpr8tUnNAJj2de2iSlohtPdO/lHQhT0vFZXVNibmabzm6Y8lgzstbHYclhWahtYFK6+RbUpNLSdfTuY5b5hwJg4BaX+hFLoA+hdj6PCsejkPKv7WTf6L8/tJWyQCSW+7Q6LPSSp7DNqYd4b+v93UfmO5SMTeiNuupjLZywyn5/Zb5tjXQwcnnbunsahuClQwUI1l+PxexWh3HFT5D2pYG6oTpKLD1yvmaXOY6KDBuJcHvayOX2/ucWkDSg1yIdOqlft76PCutHYcMjhnstfxhoyvKz5rLTpCa/f19/90Do9F/YdC5Owb94KinpIdmPT2hvljo8v6+h1+Jku+TO1q9QymKkPM6ufux5PaCecNjdRzWhUlaWhmoWnlpK0pyqek65aaZf/qFTpf2LpeUM3RRZ+vzLLR0HCWdbBVs1kan7Btz31Nv02BHBqhy/jlNb9ExlkY9XnoaJHDzR69SOYvj4J5iHs9yQl+/NOctly8Mba1xGiEXH7b4BcDWcG+8lnAELefiOEKRI2aZzsrhqadBBs+Nx7SOg9UXPW5374WXXFgYojFYIZdAcsBgnFumzDkRLtNXdBJSR0vzG6GnH4DzFR6NrnUcd2y7urQHPORSumauXDzkEqRwR+yW6fJ0JwrOb6SVnPwsDXpCTz8EJ+K9do8tOY4L/Lda7pXWcvnF+LvR6fYXD0yH+SGXYA5WOq1NC9EhyPQXJ8Gl7jF1dcXfsv/Q0w/Tcm0Lfwcl/ZGj9NpnRKRQo5VcZD197pCpgpBLUINrMdY4D054S91jddY5c56sugJCT3fFlpNLPVS47YWQS7A3Pj0RHXoaBEEQBEEQBEEQBEEQBEEQBEEQBEEQBEEQBEEQBEEQBNvxP2+fAuV58+OhAAAAAElFTkSuQmCC" alt="n = F(k_1) + F(k_2) + \ldots + F(k_m)" style="width: 199px; height: 20px;" width="199" height="20"></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: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; "><span style="">,</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: 13.6167px 7.91667px; transform-origin: 13.6167px 7.91667px; unicode-bidi: normal; "><span style="">then</span></span></div><div style="block-size: 26px; 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 13px; text-align: left; transform-origin: 384px 13px; 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: 9.70833px 7.91667px; transform-origin: 9.70833px 7.91667px; unicode-bidi: normal; "><span style="">     </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" alt="a_n = 2^{k_1-2}+2^{k_2-2}+\ldots+2^{k_m-2}" style="width: 192px; height: 26px;" width="192" height="26"></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: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; "><span style=""> </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: 357.083px 7.91667px; transform-origin: 357.083px 7.91667px; unicode-bidi: normal; "><span style="">The connection to binary numbers is that the binary representations of the Fibbinary numbers have no adjacent 1s. </span></span></div><div style="block-size: 21.25px; 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 10.625px; text-align: left; transform-origin: 384px 10.625px; 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: 42.7833px 7.91667px; transform-origin: 42.7833px 7.91667px; unicode-bidi: normal; "><span style="">For example, </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAB8AAAAoCAYAAAAG0SEsAAABJ0lEQVRYhe2WUY3DMAyGPw5lMAIlEARFMAZhUAajEAyFEA6lUAylsHuYc3Ozutmm5aaT8kl+aGz5d+y0KTQajUaj8SV6wAMj4GTNyXNfUzQCi4g7IAAzcBU71RA+S/IF6DLfVfn+VFiLXz4tPKjkbsd/KvjfpgNWSTwZMakrK9uuBFXUnhULHVWwdYqj+GNW9CKFucxm7PFtSLuOht+p4ka17kXYyhlKwlZiTaTcGU06P0MpULd8L9hz70w+b4vwbOyR+Jnb3Ca2YwnYXUKErYO7oVfiUartJHk6MGnnQewoccpnnYUH0s60TdzbtmbrR1wk7pnx/DJw263n8bv9ymWyYL81VUlfQf8NcU/FG6/ELPbSvD9BavnRK1iNdPFU+8NpNBr/gx+twmvaJTM4bAAAAABJRU5ErkJggg==" style="width: 15.5px; height: 20px;" width="15.5" height="20"></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: 148.975px 7.91667px; transform-origin: 148.975px 7.91667px; unicode-bidi: normal; "><span style=""> = 10. The Zeckendorf expansion of 7 is 2+5, or </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" alt="F(3)+F(5)" style="width: 79.5px; height: 19px;" width="79.5" height="19"></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: 17.5px 7.91667px; transform-origin: 17.5px 7.91667px; unicode-bidi: normal; "><span style="">, and </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: 9.70833px 7.91667px; transform-origin: 9.70833px 7.91667px; unicode-bidi: normal; "><span style="">     </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" alt="10=2^{3-2}+2^{5-2} = 2+8" style="width: 155px; height: 19.5px;" width="155" height="19.5"></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: 210.442px 7.91667px; transform-origin: 210.442px 7.91667px; unicode-bidi: normal; "><span style="">Also, the binary expansion of 10 is 1010, which has no adjacent 1s. </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: 83.1px 7.91667px; transform-origin: 83.1px 7.91667px; unicode-bidi: normal; "><span style="">Write a function to find the </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">n</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: 65.7333px 7.91667px; transform-origin: 65.7333px 7.91667px; unicode-bidi: normal; "><span style="">th Fibbinary number. </span></span></div></div></div>

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Nikolaos Nikolaou on 25 Oct 2020

Its a nice series of problems and i think there are more than enough untied problems for another Sequence and Series problem group. So maybe Chris its about time to curate one :)

ChrisR on 25 Oct 2020

Thanks Nikolaos. I was thinking of making a group on primes, but one on sequences and series is a good idea.

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Problem 47073. Find the nth Fibbinary number

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Problem 47073. Find the nth Fibbinary number

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