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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. 3
1, 199, 39401, 7801199, 1544598001, 305822602999, 60551330795801, 11988857674965599, 2373733268312392801, 469987198268178808999, 93055091523831091789001, 18424438134520287995413199 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Indranil Ghosh, Table of n, a(n) for n = 0..434

Tanya Khovanova, Recursive Sequences

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

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).

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

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 . [From 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)

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 *)

CROSSREFS

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

Sequence in context: A153179 A269549 A209181 * A181007 A155508 A028482

Adjacent sequences:  A097729 A097730 A097731 * A097733 A097734 A097735

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Aug 31 2004

STATUS

approved

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Last modified June 24 16:32 EDT 2017. Contains 288707 sequences.