t[n] == t[2^Floor[Log[2, n]]] + t[n - 2^Floor[Log[2, n]]] + n - 2^Floor[Log[2, n]] /; t[n] == Sum[DigitCount[k, 2, 1], {k, 0, n - 1}]

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["t", "[", "n", "]"]], "\[Equal]", RowBox[List[RowBox[List["t", "[", SuperscriptBox["2", RowBox[List["Floor", "[", RowBox[List["Log", "[", RowBox[List["2", ",", "n"]], "]"]], "]"]]], "]"]], "+", RowBox[List["t", "[", RowBox[List["n", "-", SuperscriptBox["2", RowBox[List["Floor", "[", RowBox[List["Log", "[", RowBox[List["2", ",", "n"]], "]"]], "]"]]]]], "]"]], "+", "n", "-", SuperscriptBox["2", RowBox[List["Floor", "[", RowBox[List["Log", "[", RowBox[List["2", ",", "n"]], "]"]], "]"]]]]]]], "/;", RowBox[List[RowBox[List["t", "[", "n", "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List["DigitCount", "[", RowBox[List["k", ",", "2", ",", "1"]], "]"]]]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> t </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mi> n </mi> <mo> - </mo> <msup> <mn> 2 </mn> <mrow> <mo> &#8970; </mo> <mrow> <msub> <mi> log </mi> <mn> 2 </mn> </msub> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> &#8971; </mo> </mrow> </msup> <mo> + </mo> <mrow> <mi> t </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mn> 2 </mn> <mrow> <mo> &#8970; </mo> <mrow> <msub> <mi> log </mi> <mn> 2 </mn> </msub> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> &#8971; </mo> </mrow> </msup> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> t </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <msup> <mn> 2 </mn> <mrow> <mo> &#8970; </mo> <mrow> <msub> <mi> log </mi> <mn> 2 </mn> </msub> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> &#8971; </mo> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> t </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <msubsup> <mi> s </mi> <mn> 2 </mn> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> t </ci> <ci> n </ci> </apply> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <floor /> <apply> <log /> <logbase> <cn type='integer'> 2 </cn> </logbase> <ci> n </ci> </apply> </apply> </apply> </apply> <apply> <ci> t </ci> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <floor /> <apply> <log /> <logbase> <cn type='integer'> 2 </cn> </logbase> <ci> n </ci> </apply> </apply> </apply> </apply> <apply> <ci> t </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <floor /> <apply> <log /> <logbase> <cn type='integer'> 2 </cn> </logbase> <ci> n </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> t </ci> <ci> n </ci> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> s </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <ci> k </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["t", "[", "n_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["t", "[", SuperscriptBox["2", RowBox[List["Floor", "[", RowBox[List["Log", "[", RowBox[List["2", ",", "n"]], "]"]], "]"]]], "]"]], "+", RowBox[List["t", "[", RowBox[List["n", "-", SuperscriptBox["2", RowBox[List["Floor", "[", RowBox[List["Log", "[", RowBox[List["2", ",", "n"]], "]"]], "]"]]]]], "]"]], "+", "n", "-", SuperscriptBox["2", RowBox[List["Floor", "[", RowBox[List["Log", "[", RowBox[List["2", ",", "n"]], "]"]], "]"]]]]], "/;", RowBox[List[RowBox[List["t", "[", "n", "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List["DigitCount", "[", RowBox[List["k", ",", "2", ",", "1"]], "]"]]]]]]]]]]]]

2001-10-29