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 A286035 a(n) = 3*T(A285984(n)), where T(m) is the m-th triangular number A000217(m). 4
 0, 18315, 210375, 17232775560, 197941645440, 16214284059063255, 186242898311223435, 15255987442587265956120, 175235570535035566127880, 14354328072739259079522561195, 164878797845087651200279041495, 13505958574968967401962031517525680, 155134131134672045268505114018663320 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS This sequence a(n) gives the solutions x of the 3rd degree Diophantine Bachet-Mordell equation y^2=x^3+K, with y = T(b(n))*sqrt(27*T(b(n))+1) = A286036(n) and K = (T(b(n)))^2 = A286037(n), the square of the triangular number of b(n)= A285984(n). REFERENCES V. Pletser, On some solutions of the Bachet-Mordell equation for large parameter values, to be submitted, April 2017. LINKS Vladimir Pletser, Table of n, a(n) for n = 0..1000 M.A. Bennett and A. Ghadermarzi, Data on Mordell's curve. Michael A. Bennett, Amir Ghadermarzi, Mordell's equation : a classical approach, arXiv:1311.7077 [math.NT], 2013. Eric Weisstein's World of Mathematics, Mordell Curve FORMULA Since b(n) = 264*sqrt(27*T(b(n-2))+1)+ b(n-4) = 264*sqrt(27*(b(n-2)*(b(n-2)+1)/2)+1)+ b(n-4), with b(-2)=110, b(-1)=0, b(0)=0, b(1)=110 (see A285984) and a(n) = 3*T(b(n)) (this sequence), one has : a(n) = 3*[264*sqrt(27*(b(n-2)*(b(n-2)+1)/2)+1)+ b(n-4) ]*[ 264*sqrt(27*(b(n-2)*(b(n-2)+1)/2)+1)+ b(n-4)+1]/2. Empirical g.f.: 495*x*(37 + 388*x + 37*x^2) / ((1 - x)*(1 - 970*x + x^2)*(1 + 970*x + x^2)). - Colin Barker, May 01 2017 EXAMPLE For n = 2, b(n) = 374, a(n)= 210375. For n = 3, b(n) = A285984(n) =107184. Therefore, a(n) = 3*T(b(n)) = 3*A000217(A285984(n)) = 3*A000217(107184) = 3*5744258520=17232775560. MAPLE restart: bm2:=110: bm1:=0: b0:=0: bp1:=110: print ('0, 0', '1, 18315'); for n from 2 to 1000 do b:= 264*sqrt(27* (b0^2+b0)/2+1)+bm2; a:=3*b*(b+1)/2; print(n, a); bm2:=bm1; bm1:=b0; b0:=bp1; bp1:=b; end do: CROSSREFS Cf. A285984, A286036, A286037, A285955, A006454, A000217, A006451, A081119, A054504 Sequence in context: A232142 A237737 A145822 * A217049 A251025 A203783 Adjacent sequences: A286032 A286033 A286034 * A286036 A286037 A286038 KEYWORD nonn,easy AUTHOR Vladimir Pletser, May 01 2017 STATUS approved

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Last modified June 21 21:50 EDT 2024. Contains 373559 sequences. (Running on oeis4.)