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

%I A132596 #52 Dec 31 2023 10:19:27
%S A132596 0,2,24,242,2400,23762,235224,2328482,23049600,228167522,2258625624,
%T A132596 22358088722,221322261600,2190864527282,21687323011224,
%U A132596 214682365584962,2125136332838400,21036680962799042,208241673295152024
%N A132596 X-values of solutions to the equation X*(X + 1) - 6*Y^2 = 0.
%C A132596 Or, 3*A000217(X) is a square, (3*A004189(n))^2. - _Zak Seidov_, Apr 08 2009
%C A132596 "You can find an infinite number of [different] triangular numbers such that when multiplied 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 A132596 Clifford A. Pickover, The Loom of God, Tapestries of Mathematics and Mysticism, Sterling, NY, 2009, page 33.
%H A132596 Seiichi Manyama, <a href="/A132596/b132596.txt">Table of n, a(n) for n = 0..1000</a>
%H A132596 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (11,-11,1).
%F A132596 a(n) = 10*a(n-1) - a(n-2) + 4, a(0)=0, a(1)=2.
%F A132596 a(n) = (A001079(n) - 1)/2. - _Max Alekseyev_, Nov 13 2009
%F A132596 From _R. J. Mathar_, Apr 20 2010: (Start)
%F A132596 a(n) = 11*a(n-1) - 11*a(n-2) + a(n-3) = 2*A098297(n).
%F A132596 G.f.: -2*x*(1+x) / ( (x-1)*(x^2-10*x+1) ). (End)
%F A132596 a(n) = 2*A098297(n) = (1/2)*(T(2*n,sqrt(3)) - 1), where T(n,x) is the n-th Chebyshev polynomial of the first kind. - _Peter Bala_, Dec 31 2012
%t A132596 LinearRecurrence[{11, -11, 1}, {0, 2, 24}, 19] (* _Jean-François Alcover_, Feb 26 2019 *)
%Y A132596 Cf. A007654, A001079, A000217, A098297, A108741 (Y^2), A004189 (Y).
%K A132596 nonn,easy
%O A132596 0,2
%A A132596 _Mohamed Bouhamida_, Nov 14 2007