A006454 -id:A006454 - cp's OEIS frontend

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

0, 37271025, 4917515625, 32996505944592590400, 4353432777721630310400, 29211445283110309395256454577225, 3854046352373857001854365165911025, 25860572538708927496411840821477504196161600, 3411945020082158343071838489442339152945921600, 22894081602203374655543296113789919615194083223613314225

Offset: 0

Vladimir Pletser, May 01 2017

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).

			For n = 2, b(n) = 374, a(n)= 4917515625.
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.

V. Pletser, On some solutions of the Bachet-Mordell equation for large parameter values, to be submitted, April 2017.

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

Cf. A285984, A286035, A286036, A285955, A006454, A000217, A006451, A081119, A054504.

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:

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 :
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.
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

A173202 Solutions y of the Mordell equation y^2 = x^3 - 3a^2 + 1 for a = 0,1,2, ... (solutions x are given by the sequence A000466).

Original entry on oeis.org

0, 5, 58, 207, 500, 985, 1710, 2723, 4072, 5805, 7970, 10615, 13788, 17537, 21910, 26955, 32720, 39253, 46602, 54815, 63940, 74025, 85118, 97267, 110520, 124925, 140530, 157383, 175532, 195025, 215910, 238235, 262048, 287397, 314330, 342895

Offset: 1

Michel Lagneau, Feb 12 2010

For many values of k for the equation y^2 = x^3 + k, all the solutions are known. For example, we have solutions for k=-2: (x,y) = (3,-5) and (3,5). A complete resolution for all integers k is unknown. Theorem: Let k be < -1, free of square factors, with k == 2 or 3 (mod 4). Suppose that the number of classes h(Q(sqrt(k))) is not divisible by 3. Then the equation y^2 = x^3 + k admits integer solutions if and only if k = 1 - 3a^2 or 1 - 3a^2 where a is an integer. In this case, the solutions are x = a^2 - k, y = a(a^2 + 3k) or -a(a^2 + 3k) (the first reference gives the proof of this theorem). With k = -1 - 3a^2, we obtain the solutions x = 4a^2 + 1, y = a(8a^2 + 3) or -a(8a^2 + 3). For the case k = 1 - 3a^2, we obtain the solution x = 4a^2 - 1 given by the sequence A000466.

			With a=3, x = 35 and y = 207, and then 207^2 = 35^2 - 26.

T. Apostol, Introduction to Analytic Number Theory, Springer, 1976
D. Duverney, Theorie des nombres (2e edition), Dunod, 2007, p.151

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
W. J. Ellison, F. Ellison, J. Pesek, C. E. Stall & D. S. Stall, The diophantine equation y^2 + k = x^3, J. Number Theory 4 (1972), 107-117.
Helmut Richter, Solutions of Mordell's equation y^2 = x^3 + k (solutions for 0
School of Mathematics and Statistics, University of St Andrews, Louis Joel Mordell.
Eric Weisstein's World of Mathematics, Mordell Curve.
D. J. Wright, Mordell's Equation.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Diophantine equations: see also Pellian equation: (A081233, A081234), (A081231, A082394), (A081232, A082393); Mordell equation: A053755, A173200; Diophantine equations: A006452, A006451, A006454.

I:=[0, 5, 58, 207]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jul 02 2012

for a from 0 to 100 do : z := evalf(a*(8*a^2 - 3)) : print (z) :od :

CoefficientList[Series[x*(5+38*x+5*x^2)/(1-x)^4,{x,0,40}],x] (* Vincenzo Librandi, Jul 02 2012 *)
CoefficientList[Series[E^x (5 x + 24 x^2 + 8 x^3), {x, 0, 40}], x]*Table[n!, {n, 0, 40}] (* Stefano Spezia, Dec 04 2018 *)

y = a*(8*a^2 - 3).
a(n) = sqrt(A000466(n)^3 - A080663(n)). - Artur Jasinski, Nov 26 2011
From Colin Barker, Apr 26 2012: (Start)
a(n) = 8*n^3 - 24*n^2 + 21*n - 5.
G.f.: x^2*(5 + 38*x + 5*x^2)/(1 - x)^4. (End)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jul 02 2012
E.g.f.: exp(x)*(5*x + 24*x^2 + 8*x^3). - Stefano Spezia, Dec 04 2018

A225115 Triangular numbers of the form p^k - 1, where p is a prime number and k >= 2.

Original entry on oeis.org

3, 15, 120, 528, 4095, 139128, 160554240, 812293020528, 379188080252621270252095320

Offset: 1

Alex Ratushnyak, Apr 29 2013

There are at least other 4 terms greater than a(9) that can be easily obtained by solving the Diophantine equation T(n) = p^2-1, the smallest being T(75055187665070250755513356704300447). - Giovanni Resta, Apr 29 2013

Cf. A006454, A025475.

t=0;for(n=1,10^9,t+=n; if(isprimepower(t+1)>=2,print1(t,", "))); \\ Joerg Arndt, Apr 29 2013

a(9) from Giovanni Resta, Apr 29 2013

A227480 Triangular numbers of the form p^2 - 1 where p is a prime.

Original entry on oeis.org

3, 120, 528, 139128, 160554240, 812293020528, 379188080252621270252095320, 2816640597719456759751165488439681098364673273236637971046370174350128

Offset: 1

Shreevatsa R, Jul 12 2013

nonn

			139128 is both 527*528/2 and 373^2 - 1 (and 373 is prime).

Subsequence of A006454 and of A225115.

Select[Prime[Range[100000]]^2-1,OddQ[Sqrt[8#+1]]&] (* The program generates the first six terms of the sequence. *) (* Harvey P. Dale, Sep 17 2023 *)

a(7)-a(8) from Giovanni Resta, Jul 13 2013

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

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A173202 Solutions y of the Mordell equation y^2 = x^3 - 3a^2 + 1 for a = 0,1,2, ... (solutions x are given by the sequence A000466).

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A225115 Triangular numbers of the form p^k - 1, where p is a prime number and k >= 2.

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A227480 Triangular numbers of the form p^2 - 1 where p is a prime.

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