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

0, 2, 24, 242, 2400, 23762, 235224, 2328482, 23049600, 228167522, 2258625624, 22358088722, 221322261600, 2190864527282, 21687323011224, 214682365584962, 2125136332838400, 21036680962799042, 208241673295152024

Offset

0,2

Comments

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 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

References

Clifford A. Pickover, The Loom of God, Tapestries of Mathematics and Mysticism, Sterling, NY, 2009, page 33.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (11,-11,1).

Formula

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

a(n) = (A001079(n) - 1)/2. - Max Alekseyev, Nov 13 2009

From R. J. Mathar, Apr 20 2010: (Start)

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) ). (End)

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

Mathematica

LinearRecurrence[{11, -11, 1}, {0, 2, 24}, 19] (* Jean-François Alcover, Feb 26 2019 *)

Crossrefs

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

Keyword

nonn,easy

Author

Mohamed Bouhamida, Nov 14 2007

Status

approved