A266698 x-values of solutions to the Diophantine equation x^2 - 7*y^2 = 2.

3, 45, 717, 11427, 182115, 2902413, 46256493, 737201475, 11748967107, 187246272237, 2984191388685, 47559815946723, 757972863758883, 12080006004195405, 192522123203367597, 3068273965249686147, 48899861320791610755, 779329507167416085933, 12420372253357865764173, 197946626546558436140835

Offset: 1

Author

Sture Sjöstedt, Jan 03 2016

Keywords

• nonn
• easy

Comments

A159678 gives the y-values of solutions to the Diophantine equation x^2 - 7*y^2 = 2.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..750
• Index entries for linear recurrences with constant coefficients, signature (16,-1).

Crossrefs

Cf. A041008, A077412, A157456, A159678.

Programs

Magma

[n: n in [1..2*10^7] | IsSquare((n^2-2)/7)]; // Vincenzo Librandi, Jan 06 2016

Mathematica

LinearRecurrence[{16,-1}, {3, 45}, 20 ]

PARI

lista(nn) = {print1(x = 3, ", "); print1(y = 45, ", "); for (n=2, nn, z = 16*y - x; print1(z, ", "); x = y; y = z;);} \\ Michel Marcus, Jan 05 2016

SageMath

[3*(chebyshev_U(n-1, 8) - chebyshev_U(n-2, 8)) for n in (1..30)] # G. C. Greubel, Jun 25 2022

Formula

a(1)=3, a(2)=45, a(n) = 16*a(n-1) - a(n-2).

a(n) = A041008(4n-2). - Robert Israel, Jan 05 2016

From R. J. Mathar, Jan 12 2016: (Start)

G.f.: 3*x*(1-x) / ( 1-16*x+x^2 ).

a(n) = 3*A157456(n). (End)

From G. C. Greubel, Jun 25 2022: (Start)

a(n) = 3*(ChebyshevU(n-1, 8) - ChebyshevU(n-2, 8)).

E.g.f.: exp(8*x)*(3*cosh(3*sqrt(7)*x) - sqrt(7)*sinh(3*sqrt(7)*x)) - 3. (End)