A285985 - OEIS

%I #17 Feb 16 2025 08:33:44

%S 0,9,225,14400,278784,16769025,322382025,19356600384,372051201600,

%T 22337675375625,429347532814209,25777663981977600,495466706924481600,

%U 29747402099825117409,571768151330225342025,34328476252406392070400,659819951198501829398784,39615031848108328736769225,761431651915943270106720225,45715712424248689455481003584,878691466491082103705616000000

%N Numbers a(n) = (T(b(n)))^2, where T(b(n)) is the triangular number of b(n)= A000217(b(n)) and b(n)=A006451(n). Also a(n) = parameters K of the Bachet Mordell equation y^2=x^3+K, where x= T(b(n)) = A006454(n) and y= T(b(n))* sqrt(T(b(n))+1) = A285955(n)

%C Numbers a(n) which are the square of triangular number T(b(n)), where b(n) is the sequence A006451(n) of numbers n such that T(n)+1 is a square.

%C This sequence a(n) gives also the parameters K of the 3rd degree Diophantine Bachet-Mordell equation y^2=x^3+K, with x= T(b(n)) = A006454(n) and y= T(b(n))* sqrt(T(b(n))+1) = A285955(n).

%C Also: a(n) = (A000217(A006451(n)))^2 or a(n) = A006454(n)^2.

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

%H Vladimir Pletser, <a href="/A285985/b285985.txt">Table of n, a(n) for n = 0..500</a>

%H M.A. Bennett and A. Ghadermarzi, <a href="http://www.math.ubc.ca/~bennett/BeGa-data.htm">Data on Mordell's curve</a>.

%H 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 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MordellCurve.html">Mordell Curve</a>

%F Since b(n) = 8*sqrt(T(b(n-2))+1)+ b(n-4) = 8*sqrt((b(n-2)*(b(n-2)+1)/2)+1)+ b(n-4), with b(-1)=-1, b(0)=0, b(1)=2, b(2)=5 (see A006451) and a(n) = T(b(n)) (this sequence), one has :

%F a(n) = ([8*sqrt((b(n-2)*(b(n-2)+1)/2)+1)+ b(n-4)]*[ 8*sqrt((b(n-2)*(b(n-2)+1)/2)+1)+ b(n-4)+1]/2)^2.

%F Empirical g.f.: 9*x*(1 + 24*x + 387*x^2 + 864*x^3 + 387*x^4 + 24*x^5 + x^6) / ((1 - x)*(1 - 34*x + x^2)*(1 - 6*x + x^2)*(1 + 6*x + x^2)*(1 + 34*x + x^2)). - _Colin Barker_, Apr 30 2017

%e For n=2, b(n)=5, a(n)=225.

%e For n=5, b(n)=90, a(n)= 16769025.

%e For n = 3, A006451(n) = 15. Therefore, A000217(A006451(n)) = A000217(15) = 120 and (A000217(A006451(n)))^2 = (A000217(15))^2 = (120)^2 = 14400.

%p restart: bm2:=-1: bm1:=0: bp1:=2: bp2:=5: print ('0,0','1,9','2,225'); for n from 3 to 1000 do b:= 8*sqrt((bp1^2+bp1)/2+1)+bm2; a:=(b*(b+1)/2)^2; print(n,a); bm2:=bm1; bm1:=bp1; bp1:=bp2; bp2:=b; end do:

%Y Cf. A285955, A006454, A000217, A006451, A081119, A054504.

%K nonn,easy

%O 0,2

%A _Vladimir Pletser_, Apr 30 2017