Question Number 55491 by Tinkutara last updated on 25/Feb/19
Answered by tanmay.chaudhury50@gmail.com last updated on 26/Feb/19
![trying to understand restricting under following condition 1)radius(r) taken as integer 2)centre and otherpoints lie on the circle not counted in f(r) discussion... let r=1 then intregal points are [(0,0),(1,0),(0,1),(0,−1),(−1,0)] but these red marked points not considered so f(1)=0 r=2 [(0,0),(1,0),(−1,0),(0,1),(0,−1),(1,1),(1,−1),(−1,1),(−1,−1)] x={−1,0,1} y={−1,0,1} 3c_1 ×3c_1 −1=8 r=3 x={−2,−1,0,1,2} y={−2,−1,0,1,2} 5c_1 ×5c_1 −1=24 r=4 x={−3,−2,−1,0,1,2,3} y={−3,−2,−1,0,1,2,3} 7c_1 ×7c_1 −1=49−1 [srl no] [r ] [nos of intregal points] [f(r)] 1. 1 [null] f(1)=0 2. 2 [8] f(2)=8 3 3 [24] f(3)=24 ..... .... f(r) =(1−1)+(9−1)+(25−1)+(49−1)+... =[(1^2 +3^2 +5^2 +7^2 +...(2r−1)^2 ]−r others pls check is it correct perception.. then i shall proceed further... =((r(2r+1)(2r−1))/3)−r =((r(4r^2 −1)−3r)/3) =((4r^3 −r−3r)/3) =((4r^3 −4r)/3) li_(r→∞)](Q55531.png)
$${trying}\:{to}\:\:{understand}\: \\ $$$${restricting}\:{under}\:{following}\:{condition} \\ $$$$\left.\mathrm{1}\right){radius}\left({r}\right)\:{taken}\:{as}\:{integer} \\ $$$$\left.\mathrm{2}\right){centre}\:{and}\:{otherpoints}\:\boldsymbol{{lie}}\:\boldsymbol{{on}}\:\boldsymbol{{the}}\:\boldsymbol{{circle}} \\ $$$$\boldsymbol{{not}}\:\boldsymbol{{counted}}\:\boldsymbol{{in}}\:\boldsymbol{{f}}\left(\boldsymbol{{r}}\right) \\ $$$${discussion}... \\ $$$${let}\:{r}=\mathrm{1}\:\:{then}\:{intregal}\:{points}\:{are}\:\left[\left(\mathrm{0},\mathrm{0}\right),\left(\mathrm{1},\mathrm{0}\right),\left(\mathrm{0},\mathrm{1}\right),\left(\mathrm{0},−\mathrm{1}\right),\left(−\mathrm{1},\mathrm{0}\right)\right] \\ $$$${but}\:{these}\:{red}\:{marked}\:{points}\:{not}\:{considered} \\ $$$${so}\:{f}\left(\mathrm{1}\right)=\mathrm{0} \\ $$$$ \\ $$$${r}=\mathrm{2}\:\:\:\left[\left(\mathrm{0},\mathrm{0}\right),\left(\mathrm{1},\mathrm{0}\right),\left(−\mathrm{1},\mathrm{0}\right),\left(\mathrm{0},\mathrm{1}\right),\left(\mathrm{0},−\mathrm{1}\right),\left(\mathrm{1},\mathrm{1}\right),\left(\mathrm{1},−\mathrm{1}\right),\left(−\mathrm{1},\mathrm{1}\right),\left(−\mathrm{1},−\mathrm{1}\right)\right] \\ $$$${x}=\left\{−\mathrm{1},\mathrm{0},\mathrm{1}\right\}\:\:{y}=\left\{−\mathrm{1},\mathrm{0},\mathrm{1}\right\}\:\:\mathrm{3}{c}_{\mathrm{1}} ×\mathrm{3}{c}_{\mathrm{1}} −\mathrm{1}=\mathrm{8} \\ $$$${r}=\mathrm{3}\:\:{x}=\left\{−\mathrm{2},−\mathrm{1},\mathrm{0},\mathrm{1},\mathrm{2}\right\}\:\:{y}=\left\{−\mathrm{2},−\mathrm{1},\mathrm{0},\mathrm{1},\mathrm{2}\right\} \\ $$$$\mathrm{5}{c}_{\mathrm{1}} ×\mathrm{5}{c}_{\mathrm{1}} −\mathrm{1}=\mathrm{24} \\ $$$${r}=\mathrm{4}\:\:{x}=\left\{−\mathrm{3},−\mathrm{2},−\mathrm{1},\mathrm{0},\mathrm{1},\mathrm{2},\mathrm{3}\right\} \\ $$$$\:\:\:\:{y}=\left\{−\mathrm{3},−\mathrm{2},−\mathrm{1},\mathrm{0},\mathrm{1},\mathrm{2},\mathrm{3}\right\} \\ $$$$\mathrm{7}{c}_{\mathrm{1}} ×\mathrm{7}{c}_{\mathrm{1}} −\mathrm{1}=\mathrm{49}−\mathrm{1} \\ $$$$\left[{srl}\:{no}\right]\:\:\:\left[{r}\:\:\:\right]\:\:\:\:\:\left[{nos}\:{of}\:{intregal}\:{points}\right]\:\:\:\:\left[{f}\left({r}\right)\right] \\ $$$$\:\:\:\:\:\:\mathrm{1}.\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\:\left[{null}\right]\:\:\:\:\:\:\:\:\:\:\:\:\:{f}\left(\mathrm{1}\right)=\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\mathrm{2}.\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}\:\:\:\:\:\:\:\:\:\:\:\left[\mathrm{8}\right]\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{f}\left(\mathrm{2}\right)=\mathrm{8} \\ $$$$\:\:\:\:\:\:\:\mathrm{3}\:\:\:\:\:\:\:\:\:\:\:\mathrm{3}\:\:\:\:\:\:\:\:\left[\mathrm{24}\right]\:\:\:\:\:\:\:\:\:\:\:{f}\left(\mathrm{3}\right)=\mathrm{24} \\ $$$$..... \\ $$$$.... \\ $$$${f}\left({r}\right) \\ $$$$=\left(\mathrm{1}−\mathrm{1}\right)+\left(\mathrm{9}−\mathrm{1}\right)+\left(\mathrm{25}−\mathrm{1}\right)+\left(\mathrm{49}−\mathrm{1}\right)+... \\ $$$$=\left[\left(\mathrm{1}^{\mathrm{2}} +\mathrm{3}^{\mathrm{2}} +\mathrm{5}^{\mathrm{2}} +\mathrm{7}^{\mathrm{2}} +...\left(\mathrm{2}{r}−\mathrm{1}\right)^{\mathrm{2}} \right]−{r}\right. \\ $$$${others}\:{pls}\:{check}\:{is}\:{it}\:{correct}\:{perception}.. \\ $$$${then}\:{i}\:{shall}\:{proceed}\:{further}... \\ $$$$=\frac{{r}\left(\mathrm{2}{r}+\mathrm{1}\right)\left(\mathrm{2}{r}−\mathrm{1}\right)}{\mathrm{3}}−{r} \\ $$$$=\frac{{r}\left(\mathrm{4}{r}^{\mathrm{2}} −\mathrm{1}\right)−\mathrm{3}{r}}{\mathrm{3}} \\ $$$$=\frac{\mathrm{4}{r}^{\mathrm{3}} −{r}−\mathrm{3}{r}}{\mathrm{3}} \\ $$$$=\frac{\mathrm{4}{r}^{\mathrm{3}} −\mathrm{4}{r}}{\mathrm{3}} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$\underset{{r}\rightarrow\infty} {\mathrm{li}} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$