(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 8.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 157, 7] NotebookDataLength[ 102353, 3472] NotebookOptionsPosition[ 85747, 2975] NotebookOutlinePosition[ 88959, 3057] CellTagsIndexPosition[ 88819, 3051] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[TextData[StyleBox["Computer Algebra for\nConcrete Mathematics", FontColor->RGBColor[0, 0, 1]]], "Title", CellChangeTimes->{{3.605878357218881*^9, 3.605878376926759*^9}}, TextAlignment->Center], Cell[TextData[StyleBox["DK Fundamentals Lecture", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]]], "Subtitle", CellChangeTimes->{{3.539923518941972*^9, 3.539923520274494*^9}, { 3.605878379862816*^9, 3.605878425690149*^9}, {3.6736628029044123`*^9, 3.673662819722826*^9}}, TextAlignment->Center], Cell[TextData[{ StyleBox["Peter.Paule@", "Text"], StyleBox["risc", "Text", FontColor->RGBColor[1, 0, 1]], StyleBox[".jku.at", "Text"] }], "Subtitle", CellChangeTimes->{{3.5388148295224743`*^9, 3.538814842488023*^9}}, TextAlignment->Center], Cell[TextData[StyleBox["May 31, 2016, JKU", "Subsubtitle"]], "Title", CellChangeTimes->{{3.53881485947679*^9, 3.538814887465954*^9}, { 3.568383297503654*^9, 3.568383302205538*^9}, {3.605878434241126*^9, 3.605878456781557*^9}, {3.673662823471341*^9, 3.673662837822565*^9}}, TextAlignment->Center] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell[TextData[StyleBox[" Preambel", FontColor->RGBColor[0, 0, 1]]], "Section", CellChangeTimes->{{3.538813684902931*^9, 3.538813690365329*^9}, { 3.5388137267107697`*^9, 3.538813738496101*^9}, 3.568448501397541*^9, { 3.605878513867198*^9, 3.605878534100317*^9}, {3.673668385616784*^9, 3.673668392873456*^9}}], Cell[TextData[{ "NOTE 1. 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To that end, the third-named author (DZ) offers \ a prize of one hundred ($100$) US-dollars for a short, self-contained, \ human-generated (and computer-free) proof of Gessel\[CloseCurlyQuote]s \ conjecture, not to exceed five standard pages typed in standard font.The \ longer that prize would remain unclaimed, the more (empirical) evidence we \ would have that a proof of Gessel\[CloseCurlyQuote]s conjecture is indeed \ beyond the scope of humankind.\[CloseCurlyDoubleQuote]\ \>", "Text", CellChangeTimes->{{3.539339594897356*^9, 3.539339609714857*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Fibonacci and Automatic Proving", "Section", CellChangeTimes->{{3.538813845894498*^9, 3.538813847465474*^9}}], Cell[TextData[{ StyleBox["The ", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["Fibonacci", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], StyleBox[" sequence (Leonardo Fibonacci, 1202)", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]] }], "Text", CellChangeTimes->{3.539339726302824*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"GuessRE", "[", RowBox[{ StyleBox[ RowBox[{"{", RowBox[{"1", ",", "1", ",", "2", ",", "3", ",", "5", ",", "8"}], "}"}], FontColor->RGBColor[1, 0, 0]], ",", RowBox[{"F", "[", "k", "]"}]}], "]"}]], "Input", CellChangeTimes->{{3.539339737423435*^9, 3.53933973742467*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"-", RowBox[{"F", "[", "k", "]"}]}], "-", RowBox[{"F", "[", RowBox[{"1", "+", "k"}], "]"}], "+", RowBox[{"F", "[", RowBox[{"2", "+", "k"}], "]"}]}], "\[Equal]", "0"}], ",", RowBox[{ RowBox[{"F", "[", "0", "]"}], "\[Equal]", "1"}], ",", RowBox[{ RowBox[{"F", "[", "1", "]"}], "\[Equal]", "1"}]}], "}"}], ",", "\<\"ogf\"\>"}], "}"}]], "Output", CellChangeTimes->{3.673685092560401*^9}] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"re", "=", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"-", RowBox[{"F", "[", "k", "]"}]}], "-", RowBox[{"F", "[", RowBox[{"1", "+", "k"}], "]"}], "+", RowBox[{"F", "[", RowBox[{"2", "+", "k"}], "]"}]}], "\[Equal]", "0"}], ",", RowBox[{ RowBox[{"F", "[", "0", "]"}], "\[Equal]", "1"}], ",", RowBox[{ RowBox[{"F", "[", "1", "]"}], "\[Equal]", "1"}]}], "}"}]}], ";"}]], "Input", CellChangeTimes->{{3.539339745118753*^9, 3.539339745119382*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"RSolve", "[", RowBox[{"re", ",", RowBox[{"F", "[", "k", "]"}], ",", "k"}], "]"}]], "Input", CellChangeTimes->{{3.539339751771421*^9, 3.5393397517732964`*^9}, { 3.568384035362333*^9, 3.568384036704128*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{"F", "[", "k", "]"}], "\[Rule]", RowBox[{ FractionBox["1", "2"], " ", RowBox[{"(", RowBox[{ RowBox[{"Fibonacci", "[", "k", "]"}], "+", RowBox[{"LucasL", "[", "k", "]"}]}], ")"}]}]}], "}"}], "}"}]], "Output",\ CellChangeTimes->{3.673685114594425*^9}] }, Open ]], Cell["\<\ Is this a correct solution of the Fibonacci difference equations? 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The grade you obtain on this part of the \[OpenCurlyDoubleQuote]DK \ Fundamentals\[CloseCurlyDoubleQuote] depends on how you work out these \ Homework Problems. Submit your solutions either in hardcopy or in electronic \ form. Deadline: basically no deadline. (But I recommed not to delay for too long;)\ \>", "Text", CellChangeTimes->{{3.568714215425003*^9, 3.568714418067738*^9}, { 3.568717029312062*^9, 3.568717029796621*^9}}], Cell[CellGroupData[{ Cell["Homework Problem 1", "Subsection", CellChangeTimes->{{3.53881295962811*^9, 3.538812966560772*^9}, { 3.568714068360773*^9, 3.56871407596576*^9}, {3.568714166159519*^9, 3.56871417053851*^9}, {3.568714204727184*^9, 3.568714209822743*^9}, { 3.568714453349378*^9, 3.568714455758183*^9}}], Cell[TextData[{ "Consider ", StyleBox["Mathematica\[CloseCurlyQuote]", FontSlant->"Italic"], "s output to \[OpenCurlyDoubleQuote]RSolve\[CloseCurlyDoubleQuote]. \ Question: Is it correct?" }], "Text", CellChangeTimes->{{3.568714788456022*^9, 3.56871490861074*^9}, { 3.568717041267195*^9, 3.568717045914532*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"re", "=", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"-", RowBox[{"F", "[", "k", "]"}]}], "-", RowBox[{"F", "[", RowBox[{"1", "+", "k"}], "]"}], "+", RowBox[{"F", "[", 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the \[OpenCurlyDoubleQuote]Automatic Proof \[OpenCurlyDoubleQuote] \ (given above) of the \[OpenCurlyDoubleQuote]Binomial Theorem\ \[CloseCurlyDoubleQuote]\ \>", "Text", CellChangeTimes->{{3.568714788456022*^9, 3.56871490861074*^9}, { 3.568715091402282*^9, 3.56871512250201*^9}, {3.568715152750206*^9, 3.568715153839347*^9}, {3.568715186912631*^9, 3.568715188241767*^9}, { 3.568715236561785*^9, 3.568715249226001*^9}, {3.6736699407063828`*^9, 3.6736699439396753`*^9}}], Cell[BoxData[ RowBox[{" ", RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", "1", ")"}], " ", RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"k", "=", "0"}], "n"], RowBox[{ RowBox[{"(", GridBox[{ {"n"}, {"k"} }], ")"}], " ", SuperscriptBox["x", "k"], SuperscriptBox["y", RowBox[{"n", "-", "k"}]]}]}]}], "=", SuperscriptBox[ RowBox[{"(", RowBox[{"x", "+", "y"}], ")"}], "n"]}], ",", " ", RowBox[{ RowBox[{"n", "\[GreaterEqual]", " ", "0"}], ":"}]}]}]], "Input", CellChangeTimes->{{3.568384673232225*^9, 3.568384716063074*^9}, { 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