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

0, 110, 374, 107184, 363264, 103968854, 352366190, 100849681680, 341794841520, 97824087261230, 331540643908694, 94889263793711904, 321594082796592144, 92042488055813286134, 311945928772050471470, 89281118524875093838560, 302587229314806160734240, 86602592926640785210117550

Offset: 0

Author

Vladimir Pletser, May 01 2017

Keywords

• nonn
• easy

Comments

Numbers a(n) that make sqrt(27*T(a(n))+1) an integer.

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).

Examples

			k = 110 is a term because 27*(T(110) + 1) = 27 * (110*111/2 + 1) is a square. - _David A. Corneth_, May 02 2017
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.
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.
		

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 and Amir Ghadermarzi, Mordell's equation : a classical approach, arXiv:1311.7077 [math.NT], 2013.
• Eric Weisstein's World of Mathematics, Mordell Curve

Crossrefs

Cf. A286035, A286036, A286037, A285955, A006454, A000217, A006451, A081119, A054504.

Programs

Maple

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:

Mathematica

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 *)

PARI

is(n) = issquare(27*binomial(n+1, 2)+1) \\ David A. Corneth, May 02 2017

Formula

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.

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

a(n) = 485*a(n-2)+242+66*sqrt(54*a(n-2)^2+54*a(n-2)+4). - Robert Israel, May 03 2017