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A285984 Numbers k such that 27*T(k)+1 is a square, where T(m) is the m-th triangular number A000217(m).

%I A285984 #25 Feb 16 2025 08:33:44
%S A285984 0,110,374,107184,363264,103968854,352366190,100849681680,
%T A285984 341794841520,97824087261230,331540643908694,94889263793711904,
%U A285984 321594082796592144,92042488055813286134,311945928772050471470,89281118524875093838560,302587229314806160734240,86602592926640785210117550
%N A285984 Numbers k such that 27*T(k)+1 is a square, where T(m) is the m-th triangular number A000217(m).
%C A285984 Numbers a(n) that make sqrt(27*T(a(n))+1) an integer.
%C A285984 This sequence a(n) gives also the indices of the triangular numbers T(a(n)) such that the 3rd degree Diophantine Bachet-Mordell equation y^2 = x^3+K holds with x = 3*T(a(n)) = A286035(n), y = T(a(n))* sqrt(27*T(a(n))+1) = A286036(n) and K = T(a(n))^2 = A286037(n).
%D A285984 V. Pletser, On some solutions of the Bachet-Mordell equation for large parameter values, to be submitted, April 2017.
%H A285984 Vladimir Pletser, <a href="/A285984/b285984.txt">Table of n, a(n) for n = 0..1000</a>
%H A285984 M.A. Bennett and A. Ghadermarzi, <a href="http://www.math.ubc.ca/~bennett/BeGa-data.htm">Data on Mordell's curve</a>.
%H A285984 Michael A. Bennett and Amir Ghadermarzi, <a href="https://arxiv.org/abs/1311.7077">Mordell's equation : a classical approach</a>, arXiv:1311.7077 [math.NT], 2013.
%H A285984 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MordellCurve.html">Mordell Curve</a>
%F A285984 a(n) = 264*sqrt(27*T(a(n-2))+1)+ a(n-4) = 264*sqrt(27*(a(n-2)*(a(n-2)+1)/2)+1)+ a(n-4), with a(-2)=110, a(-1)=0, a(0)=0, a(1)=110.
%F A285984 Empirical g.f.: 22*x*(5 + 12*x + 5*x^2) / ((1 - x)*(1 - 970*x^2 + x^4)). - _Colin Barker_, May 01 2017, verified by _Robert Israel_, May 03 2017
%F A285984 a(n) = 485*a(n-2)+242+66*sqrt(54*a(n-2)^2+54*a(n-2)+4). - _Robert Israel_, May 03 2017
%e A285984 k = 110 is a term because 27*(T(110) + 1) = 27 * (110*111/2 + 1) is a square. - _David A. Corneth_, May 02 2017
%e A285984 For n = 2, a(2) = 264*sqrt(27*(a(0)*(a(0)+1)/2)+1)+ a(-2) = 264*sqrt(27*(0*(0+1)/2)+1) + 110 = 374.
%e A285984 For n = 6, a(6) = 264*sqrt(27*(a(4)*(a(4)+1)/2)+1)+ a(2) = 264*sqrt(27*(363264*(363264+1)/2)+1) + 374 = 352366190.
%p A285984 restart: am2:=110: am1:=0: a0:=0: ap1:=110: print ('0,0','1,110'); for n from 2 to 1000 do a:= 264*sqrt(27* (a0^2+a0)/2+1)+am2; print(n,a); am2:=am1; am1:=a0; a0:=ap1; ap1:=a; end do:
%t A285984 nxt[{a_,b_}]:={b,485*a+242+66*Sqrt[54a^2+54*a+4]}; NestList[nxt,{0,110},20][[All,1]] (* _Harvey P. Dale_, May 30 2018 *)
%o A285984 (PARI) is(n) = issquare(27*binomial(n+1, 2)+1) \\ _David A. Corneth_, May 02 2017
%Y A285984 Cf. A286035, A286036, A286037, A285955, A006454, A000217, A006451, A081119, A054504.
%K A285984 nonn,easy
%O A285984 0,2
%A A285984 _Vladimir Pletser_, May 01 2017