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Question Number 118650 by Hassen_Timol last updated on 18/Oct/20

$$\mathrm{What}\mathrm{is}\mathrm{the}\left(\mathrm{explicit}\right)\mathrm{formula}\mathrm{of}\mathrm{this}\mathrm{sequence}?$$$$19,151,1021,50389$$

Answered by MJS_new last updated on 18/Oct/20

$$\mathrm{zillions}\mathrm{of}\mathrm{possible}\mathrm{formulas}$$$$\mathrm{i}.\mathrm{e}.$$$$a_n=7960n^3-47391n^2+86585n-47135$$$$b_n=\frac{147601927861n^3}{694834860n^3-6153076035n^2+13646831254n-420067560}$$

Commented by Hassen_Timol last updated on 18/Oct/20

$$\mathrm{Thanks}\mathrm{a}\mathrm{lot}...\mathrm{Do}\mathrm{you}\mathrm{know}\mathrm{how}\mathrm{to}\mathrm{find}?$$

Commented by MJS_new last updated on 19/Oct/20

$$\mathrm{the}\mathrm{easiest}\mathrm{and}\mathrm{most}\mathrm{usual}\mathrm{is}\mathrm{to}\mathrm{find}\mathrm{a}$$$$\mathrm{polynome}$$$${\mathrm{we}}^{\prime }\mathrm{ve}\mathrm{got}4\mathrm{numbers}{y}_1,y_2,y_3,y_4$$$$\Rightarrow \mathrm{polynome}\mathrm{of}\mathrm{degree}3$$$$y_j={aj}^3+{bj}^2+cj+d,j=1,2,3,4$$$$\mathrm{this}\mathrm{leads}\mathrm{to}\mathrm{a}\mathrm{linear}\mathrm{system}\mathrm{if}4\mathrm{equations}$$$$\mathrm{in}4\mathrm{unknowns}a,b,c,d.\mathrm{solve}\mathrm{it}\mathrm{and}{\mathrm{you}}^{\prime }\mathrm{re}$$$$\mathrm{done}.\mathrm{in}\mathrm{this}\mathrm{case}\mathrm{it}\mathrm{leads}\mathrm{to}\mathrm{my}\mathrm{first}\mathrm{solution}$$

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