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

0,2

Comments

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

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).

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

Example

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

Mathematica

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

Prog

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

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.

Sequence in context: A324436 A209181 A298251 * A305413 A181007 A155508

Adjacent sequences: A097729 A097730 A097731 * A097733 A097734 A097735

Keyword

nonn,easy

Author

Wolfdieter Lang, Aug 31 2004

Status

approved