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).

0, 2478630, 96492000, 2262209634604920, 88065491686677120, 2064651070850763887750940, 80374740223699340246041830, 1884345278651963087653858708518360, 73355621393690297028946986338029560, 1719785575058362227821108881720941727234290, 66949481579385248741161156467886515267346140

Offset

0,2

Comments

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.

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) = T(b(n))*sqrt(27*T(b(n))+1) (this sequence), one has :

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).

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

Example

For n = 2, b(n) = 374, a(n)= 96492000.
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.

Maple

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:

Crossrefs

Cf. A285984, A286035, A286037, A285955, A006454, A000217, A006451, A081119, A054504.

Sequence in context: A254727 A230015 A180705 * A215814 A114659 A179733

Adjacent sequences: A286033 A286034 A286035 * A286037 A286038 A286039

Keyword

nonn,easy

Author

Vladimir Pletser, May 01 2017

Status

approved