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A132596 X-values of solutions to the equation X*(X + 1) - 6*Y^2 = 0. 10

%I

%S 0,2,24,242,2400,23762,235224,2328482,23049600,228167522,2258625624,

%T 22358088722,221322261600,2190864527282,21687323011224,

%U 214682365584962,2125136332838400,21036680962799042,208241673295152024

%N X-values of solutions to the equation X*(X + 1) - 6*Y^2 = 0.

%C Twice A098297. [_Peter Bala_, Dec 31 2012]

%C Or, 3*A000217(X) is a square. [_Zak Seidov_, Apr 08 2009]

%C "You can find an infinite number of [different] triangular numbers such that when multipled together form a square number. For example, for every triangular number, T_n, there are an infinite number of other triangular numbers, T_m, such that T_n*T_m is a square. For example, T_2 * T_24 = 30^2." Pickover. [_Robert G. Wilson v_, Apr 01 2010]

%D Clifford A. Pickover, The Loom of God, Tapestries of Mathematics and Mysticism, Sterling, NY, 2009, page 33. [From _Robert G. Wilson v_, Apr 01 2010]

%H Seiichi Manyama, <a href="/A132596/b132596.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (11,-11,1). [_R. J. Mathar_, Apr 20 2010]

%F a(n) = 10*a(n-1) - a(n-2) + 4, a(0)=0, a(1)=2.

%F a(n) = -1/2+(1/4)*(5+2*sqrt(6))^n+(1/4)*(5-2*sqrt(6))^n, with n>=0. - _Paolo P. Lava_, Jul 15 2008

%F a(n) = (A001079(n) - 1)/2. [_Max Alekseyev_, Nov 13 2009]

%F a(n) = 11*a(n-1) -11*a(n-2) +a(n-3) = 2*A098297(n). G.f.: -2*x*(1+x) / ( (x-1)*(x^2-10*x+1) ). [_R. J. Mathar_, Apr 20 2010]

%F a(n) = 2*A098297(n) = 1/2*(T(2*n,sqrt(3)) - 1), T(n,x) the n-th Chebyshev polynomial of the first kind. - _Peter Bala_, Dec 31 2012

%t LinearRecurrence[{11, -11, 1}, {0, 2, 24}, 19] (* _Jean-Fran├žois Alcover_, Feb 26 2019 *)

%Y Cf. A007654, A001079, A000217, A098297.

%K nonn

%O 0,2

%A _Mohamed Bouhamida_, Nov 14 2007

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Last modified May 31 22:45 EDT 2020. Contains 334756 sequences. (Running on oeis4.)