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A286036 a(n) is the solution y to the Bachet Mordell equation y^2=x^3+K, with x = 3*T(b(n)) and K = (T(b(n)))^2, where T(b(n)) is the triangular number of b(n)= A285984(n).

%I A286036 #18 Feb 16 2025 08:33:44
%S A286036 0,2478630,96492000,2262209634604920,88065491686677120,
%T A286036 2064651070850763887750940,80374740223699340246041830,
%U A286036 1884345278651963087653858708518360,73355621393690297028946986338029560,1719785575058362227821108881720941727234290,66949481579385248741161156467886515267346140
%N A286036 a(n) is the solution y to the Bachet Mordell equation y^2=x^3+K, with x = 3*T(b(n)) and K = (T(b(n)))^2, where T(b(n)) is the triangular number of b(n)= A285984(n).
%C A286036 a(n) is the producs of the triangular number T(b(n)) and the square root of 27 times this triangular number plus one, sqrt(27*T(b(n))+1), where b(n) is the sequence A285984(n) of numbers n such that (27*T(n)+1) is a square.
%D A286036 V. Pletser, On some solutions of the Bachet-Mordell equation for large parameter values, to be submitted, April 2017.
%H A286036 Vladimir Pletser, <a href="/A286036/b286036.txt">Table of n, a(n) for n = 0..1000</a>
%H A286036 M.A. Bennett and A. Ghadermarzi, <a href="http://www.math.ubc.ca/~bennett/BeGa-data.htm">Data on Mordell's curve</a>.
%H A286036 Michael A. Bennett, Amir Ghadermarzi, <a href="https://arxiv.org/abs/1311.7077">Mordell's equation : a classical approach</a>, arXiv:1311.7077 [math.NT], 2013.
%H A286036 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MordellCurve.html">Mordell Curve</a>
%F A286036 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) = T(b(n))*sqrt(27*T(b(n))+1) (this sequence), one has :
%F A286036 a(n) = ([264*sqrt(27*T(b(n-2))+1)+ b(n-4)]*[ 264*sqrt(27*T(b(n-2))+1)+ b(n-4)+1]/2) *sqrt(27*([264*sqrt(27*T(b(n-2))+1)+ b(n-4)]*[ 264*sqrt(27*T(b(n-2))+1)+ b(n-4)+1]/2)+1).
%F A286036 Empirical g.f.: 330*x*(1 + x)*(7511 + 284889*x + 108094375*x^2 + 284889*x^3 + 7511*x^4) / ((1 - 912670090*x^2 + x^4)*(1 - 970*x^2 + x^4)). - _Colin Barker_, May 01 2017
%e A286036 For n = 2, b(n) = 374, a(n)= 96492000.
%e A286036 For n = 3, b(n) = A285984(n) =107184. Therefore, a(n) = T(b(n))* sqrt(27*T(b(n))+1) = A000217(A285984(n))* sqrt(27*A000217(A285984(n))+1) = A000217(107184)* sqrt(27*A000217(107184)+1) =5744258520* sqrt(27*5744258520 +1) = 2262209634604920.
%p A286036 restart: bm2:=110: bm1:=0: b0:=0: bp1:=110: print ('0,0','1,2478630’); for n from 2 to 1000 do b:= 264*sqrt(27* (b0^2+b0)/2+1)+bm2; T:=b*(b+1)/2; a:= T*sqrt(27*T+1); print(n,a); bm2:=bm1; bm1:=b0; b0:=bp1; bp1:=b; end do:
%Y A286036 Cf. A285984, A286035, A286037, A285955, A006454, A000217, A006451, A081119, A054504.
%K A286036 nonn,easy
%O A286036 0,2
%A A286036 _Vladimir Pletser_, May 01 2017