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A132596
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X-values of solutions to the equation X*(X + 1) - 6*Y^2 = 0.
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10
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0, 2, 24, 242, 2400, 23762, 235224, 2328482, 23049600, 228167522, 2258625624, 22358088722, 221322261600, 2190864527282, 21687323011224, 214682365584962, 2125136332838400, 21036680962799042, 208241673295152024
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OFFSET
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0,2
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COMMENTS
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Or, 3*A000217(X) is a square, (3*A004189(n))^2. [Zak Seidov, Apr 08 2009]
"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]
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REFERENCES
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Clifford A. Pickover, The Loom of God, Tapestries of Mathematics and Mysticism, Sterling, NY, 2009, page 33.
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LINKS
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Seiichi Manyama, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11,-11,1).
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FORMULA
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a(n) = 10*a(n-1) - a(n-2) + 4, a(0)=0, a(1)=2.
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
a(n) = (A001079(n) - 1)/2. [Max Alekseyev, Nov 13 2009]
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]
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
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MATHEMATICA
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LinearRecurrence[{11, -11, 1}, {0, 2, 24}, 19] (* Jean-François Alcover, Feb 26 2019 *)
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CROSSREFS
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Cf. A007654, A001079, A000217, A098297, A108741 (Y^2), A004189 (Y).
Sequence in context: A228619 A252764 A215929 * A099669 A019520 A300400
Adjacent sequences: A132593 A132594 A132595 * A132597 A132598 A132599
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KEYWORD
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nonn,easy
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AUTHOR
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Mohamed Bouhamida, Nov 14 2007
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STATUS
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approved
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