A097732 Pell equation solutions (7*a(n))^2 - 2*(5*b(n))^2 = -1 with b(n):=A097733(n), n >= 0. Note that D=50=2*5^2 is not squarefree.

1, 199, 39401, 7801199, 1544598001, 305822602999, 60551330795801, 11988857674965599, 2373733268312392801, 469987198268178808999, 93055091523831091789001, 18424438134520287995413199, 3647945695543493192000024401, 722274823279477131728009418199

Offset: 0

Author

Wolfdieter Lang, Aug 31 2004

Keywords

• nonn
• easy

Comments

Also numbers k such that (7*k+1)^2 + (7*k-1)^2 is a square. - Bruno Berselli, Oct 11 2019

Examples

			(x,y) = (7,1), (1393,197), (275807,39005), ... give the positive integer solutions to x^2 - 50*y^2 =-1.
		

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..434
• Christian Aebi and Grant Cairns, Lattice equable quadrilaterals III: tangential and extangential cases, Integers (2023) Vol. 23, #A48.
• Tanya Khovanova, Recursive Sequences
• Giovanni Lucca, Integer Sequences and Circle Chains Inside a Hyperbola, Forum Geometricorum (2019) Vol. 19, 11-16.
• H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
• H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume.
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (198,-1).

Crossrefs

Cf. A097731 for S(n, 2*99), A100047.

Cf. similar sequences of the type (1/k)*sinh((2*n+1)*arcsinh(k)) listed in A097775.

Programs

Mathematica

LinearRecurrence[{198, -1}, {1, 199}, 12] (* Ray Chandler, Aug 11 2015 *)

PARI

x='x+O('x^99); Vec((1+x)/(1-2*99*x+x^2)) \\ Altug Alkan, Apr 05 2018

Formula

G.f.: (1 + x)/(1 - 2*99*x + x^2).

a(n) = S(n, 2*99) + S(n-1, 2*99) = S(2*n, 10*sqrt(2)), with Chebyshev polynomials of the 2nd kind. See A049310 for the triangle of S(n, x)= U(n, x/2) coefficients. S(-1, x) := 0 =: U(-1, x).

a(n) = ((-1)^n)*T(2*n+1, 7*i)/(7*i) with the imaginary unit i and Chebyshev polynomials of the first kind. See the T-triangle A053120.

a(n) = 198*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=199. - Philippe Deléham, Nov 18 2008

From Peter Bala, Mar 23 2015: (Start)

a(n) = ( Pell(6*n + 6 - 2*k) + Pell(6*n + 2*k) )/( Pell(6 - 2*k) + Pell(2*k) ), for k an arbitrary integer.

a(n) = ( Pell(6*n + 6 - 2*k - 1) - Pell(6*n + 2*k + 1) )/( Pell(6 - 2*k - 1) - Pell(2*k + 1) ), for k an arbitrary integer, k != 1.

The aerated sequence (b(n))n>=1 = [1, 0, 199, 0, 39401, 0, 7801199, 0, ...] is a fourth-order linear divisibility sequence; that is, if n | m then b(n) | b(m). It is the case P1 = 0, P2 = -196, Q = -1 of the 3-parameter family of divisibility sequences found by Williams and Guy. See A100047 for the connection with Chebyshev polynomials. (End)

a(n) = (1/7)*sinh((2*n + 1)*arcsinh(7)). - Bruno Berselli, Apr 03 2018