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A286037 a(n) = T(A285984(n))^2, where T(m) is the m-th triangular number A000217(m).

%I A286037 #21 Feb 16 2025 08:33:44
%S A286037 0,37271025,4917515625,32996505944592590400,4353432777721630310400,
%T A286037 29211445283110309395256454577225,3854046352373857001854365165911025,
%U A286037 25860572538708927496411840821477504196161600,3411945020082158343071838489442339152945921600,22894081602203374655543296113789919615194083223613314225
%N A286037 a(n) = T(A285984(n))^2, where T(m) is the m-th triangular number A000217(m).
%C A286037 a(n) =(T(b(n)))^2, parameters K=a(n) of the Bachet Mordell equation y^2=x^3+K, with x= 3*T(b(n)) and y= T(b(n))*sqrt(27*T(b(n))+1), where T(b(n)) is the triangular number of b(n)= A285984(n).
%D A286037 V. Pletser, On some solutions of the Bachet-Mordell equation for large parameter values, to be submitted, April 2017.
%H A286037 Vladimir Pletser, <a href="/A286037/b286037.txt">Table of n, a(n) for n = 0..1000</a>
%H A286037 M.A. Bennett and A. Ghadermarzi, <a href="http://www.math.ubc.ca/~bennett/BeGa-data.htm">Data on Mordell's curve</a>.
%H A286037 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 A286037 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MordellCurve.html">Mordell Curve</a>
%F A286037 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)))^2 (this sequence), one has :
%F A286037 a(n) = ([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)^2.
%F A286037 Empirical g.f.: 27225*x*(1369 + 179256*x + 30879367019*x^2 + 168661970400*x^3 + 30879367019*x^4 + 179256*x^5 + 1369*x^6) / ((1 - x)*(1 - 940898*x + x^2)*(1 - 970*x + x^2)*(1 + 970*x + x^2)*(1 + 940898*x + x^2)). - _Colin Barker_, May 01 2017
%e A286037 For n = 2, b(n) = 374, a(n)= 4917515625.
%e A286037 For n = 3, b(n) = A285984(n) =107184. Therefore, a(n) = (T(b(n)))^2 = (A000217(A285984(n)))^2 = (A000217(107184))^2 = (5744258520)^2=32996505944592590400.
%p A286037 restart: bm2:=110: bm1:=0: b0:=0: bp1:=110: print ('0,0','1,4917515625’); for n from 2 to 1000 do b:= 264*sqrt(27*(b0^2+b0)/2+1)+bm2; a:=(b*(b+1)/2)^2; print(n,a); bm2:=bm1; bm1:=b0; b0:=bp1; bp1:=b; end do:
%Y A286037 Cf. A285984, A286035, A286036, A285955, A006454, A000217, A006451, A081119, A054504.
%K A286037 nonn,easy
%O A286037 0,2
%A A286037 _Vladimir Pletser_, May 01 2017