n=100;(*円の数*)
r=1.5;
list={{Cos[t]+r^-1*Cos[3*t],Sin[t]+r^-1*Sin[3*t]}};(*円の中心の座標*)
For[i=4,i<=n,i++,add={r^(-i+2)*Cos[(2*(i-2)+1)*t],r^(-i+2)*Sin[(2*(i-2)+1)*t]};
AppendTo[list,Last[list]+add]];(*円周上の点の座標かつ次の円の中心の座標*)
circ1=Circle[{0,0},1];
circ2=Circle[{Cos[t],Sin[t]},r^-1];
circ3=Circle[{Cos[t]+r^-1*Cos[3*t],Sin[t]+r^-1*Sin[3*t]},r^-2];
circ={};
For[i=4,i<=n,i++,AppendTo[circ,Circle[Part[list,i-2],r^(-i+1)]]];
AppendTo[dots,Point[Last[list]+{r^(-n+1)*Cos[(2*(n-1)+1)*t],r^(-n+1)*Sin[(2*(n-1)+1)*t]}]];(*一番最後の円の円周上の点の座標*)
gifs={};
Do[
If[t==0,dots={PointSize[0.01]}];
AppendTo[dots,Point[Last[list]+{r^(-n+1)*Cos[(2*(n-1)+1)*t],r^(-n+1)*Sin[(2*(n-1)+1)*t]}]];
gif=Graphics[{circ1,circ2,circ3,circ,dots},{PlotRange->{{-4,4},{-4,4}},AspectRatio->1}];AppendTo[gifs,gif],{t,0,2 Pi,Pi/60}];
Directory[]
Export["test.gif",gifs]

p2=Animate[Plot[Sin[c+t]+0.6666666666666666` Sin[3 (c+t)]+0.4444444444444444` Sin[5 (c+t)]+0.2962962962962963` Sin[7 (c+t)]+0.19753086419753085` Sin[9 (c+t)]+0.13168724279835392` Sin[11 (c+t)]+0.0877914951989026` Sin[13 (c+t)]+0.05852766346593507` Sin[15 (c+t)]+0.03901844231062338` Sin[17 (c+t)]+0.02601229487374892` Sin[19 (c+t)]+0.017341529915832612` Sin[21 (c+t)]+0.011561019943888409` Sin[23 (c+t)]+0.0077073466292589396` Sin[25 (c+t)]+0.005138231086172626` Sin[27 (c+t)]+0.0034254873907817508` Sin[29 (c+t)]+0.002283658260521167` Sin[31 (c+t)]+0.0015224388403474447` Sin[33 (c+t)]+0.0010149592268982965` Sin[35 (c+t)]+0.0006766394845988643` Sin[37 (c+t)]+0.0004510929897325762` Sin[39 (c+t)]+0.0003007286598217175` Sin[41 (c+t)]+0.00020048577321447834` Sin[43 (c+t)]+0.00013365718214298556` Sin[45 (c+t)]+0.00008910478809532371` Sin[47 (c+t)]+0.000059403192063549134` Sin[49 (c+t)]+0.00003960212804236609` Sin[51 (c+t)]+0.000026401418694910726` Sin[53 (c+t)]+0.00001760094579660715` Sin[55 (c+t)]+0.000011733963864404767` Sin[57 (c+t)]+7.822642576269845`*^-6 Sin[59 (c+t)]+5.2150950508465636`*^-6 Sin[61 (c+t)]+3.476730033897709`*^-6 Sin[63 (c+t)]+2.3178200225984725`*^-6 Sin[65 (c+t)]+1.5452133483989817`*^-6 Sin[67 (c+t)]+1.0301422322659878`*^-6 Sin[69 (c+t)]+6.867614881773253`*^-7 Sin[71 (c+t)]+4.5784099211821676`*^-7 Sin[73 (c+t)]+3.052273280788112`*^-7 Sin[75 (c+t)]+2.0348488538587415`*^-7 Sin[77 (c+t)]+1.3565659025724942`*^-7 Sin[79 (c+t)]+9.043772683816628`*^-8 Sin[81 (c+t)]+6.029181789211086`*^-8 Sin[83 (c+t)]+4.019454526140723`*^-8 Sin[85 (c+t)]+2.6796363507604823`*^-8 Sin[87 (c+t)]+1.7864242338403216`*^-8 Sin[89 (c+t)]+1.1909494892268811`*^-8 Sin[91 (c+t)]+7.93966326151254`*^-9 Sin[93 (c+t)]+5.29310884100836`*^-9 Sin[95 (c+t)]+3.5287392273389067`*^-9 Sin[97 (c+t)]+2.352492818225938`*^-9 Sin[99 (c+t)]+1.5683285454839587`*^-9 Sin[101 (c+t)]+1.0455523636559724`*^-9 Sin[103 (c+t)]+6.970349091039816`*^-10 Sin[105 (c+t)]+4.646899394026544`*^-10 Sin[107 (c+t)]+3.097932929351029`*^-10 Sin[109 (c+t)]+2.0652886195673526`*^-10 Sin[111 (c+t)]+1.3768590797115687`*^-10 Sin[113 (c+t)]+9.179060531410458`*^-11 Sin[115 (c+t)]+6.119373687606971`*^-11 Sin[117 (c+t)]+4.0795824584046475`*^-11 Sin[119 (c+t)]+2.7197216389364318`*^-11 Sin[121 (c+t)]+1.8131477592909543`*^-11 Sin[123 (c+t)]+1.2087651728606363`*^-11 Sin[125 (c+t)]+8.058434485737576`*^-12 Sin[127 (c+t)]+5.372289657158384`*^-12 Sin[129 (c+t)]+3.581526438105589`*^-12 Sin[131 (c+t)]+2.387684292070393`*^-12 Sin[133 (c+t)]+1.5917895280469286`*^-12 Sin[135 (c+t)]+1.0611930186979524`*^-12 Sin[137 (c+t)]+7.074620124653016`*^-13 Sin[139 (c+t)]+4.716413416435345`*^-13 Sin[141 (c+t)]+3.144275610956896`*^-13 Sin[143 (c+t)]+2.0961837406379305`*^-13 Sin[145 (c+t)]+1.3974558270919536`*^-13 Sin[147 (c+t)]+9.316372180613023`*^-14 Sin[149 (c+t)]+6.21091478707535`*^-14 Sin[151 (c+t)]+4.1406098580502336`*^-14 Sin[153 (c+t)]+2.7604065720334894`*^-14 Sin[155 (c+t)]+1.840271048022326`*^-14 Sin[157 (c+t)]+1.2268473653482174`*^-14 Sin[159 (c+t)]+8.178982435654782`*^-15 Sin[161 (c+t)]+5.4526549571031874`*^-15 Sin[163 (c+t)]+3.635103304735458`*^-15 Sin[165 (c+t)]+2.4234022031569723`*^-15 Sin[167 (c+t)]+1.6156014687713148`*^-15 Sin[169 (c+t)]+1.0770676458475433`*^-15 Sin[171 (c+t)]+7.180450972316955`*^-16 Sin[173 (c+t)]+4.78696731487797`*^-16 Sin[175 (c+t)]+3.1913115432519804`*^-16 Sin[177 (c+t)]+2.1275410288346537`*^-16 Sin[179 (c+t)]+1.4183606858897692`*^-16 Sin[181 (c+t)]+9.455737905931795`*^-17 Sin[183 (c+t)]+6.303825270621194`*^-17 Sin[185 (c+t)]+4.20255018041413`*^-17 Sin[187 (c+t)]+2.8017001202760865`*^-17 Sin[189 (c+t)]+1.8678000801840575`*^-17 Sin[191 (c+t)]+1.2452000534560385`*^-17 Sin[193 (c+t)]+8.301333689706923`*^-18 Sin[195 (c+t)]+5.534222459804615`*^-18 Sin[197 (c+t)]+3.689481639869743`*^-18 Sin[199 (c+t)],{t,0,4 Pi},PlotRange->2],{c,0,2 Pi,Appearance->"Open"},AnimationRunning->False]