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A097733 Pell equation solutions (7*b(n))^2 - 2*(5*a(n))^2 = -1 with b(n):=A097732(n), n>=0. Note that D=50=2*5^2 is not squarefree. 6
1, 197, 39005, 7722793, 1529074009, 302748930989, 59942759261813, 11868363584907985, 2349876047052519217, 465263588952813896981, 92119840736610099083021, 18239263202259846804541177 (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) = T(2*n+1, 5*sqrt(2))/(5*sqrt(2)), with Chebyshev polynomials of the 2nd and first kind. See A049310 for the triangle of S(n, x)= U(n, x/2) coefficients. S(-1, x) := 0 =: U(-1, x); and A053120 for the T-triangle.

a(n) = ((-1)^n)*S(2*n, 14*I) with the imaginary unit I and Chebyshev polynomials S(n, x) with coefficients shown in A049310.

G.f.: (1-x)/(1-198*x+x^2).

a(n) = 198*a(n-1)-a(n-2), n>1 ; a(0)=1, a(1)=197 . [From Philippe Deléham, Nov 18 2008]

a(n) = k^n+k^(-n)-a(n-1) = A003499(3n)-a(n-1), where k = (sqrt(2)+1)^6 = 99+70*sqrt(2) and a(0)=1. - Charles L. Hohn, Apr 05 2011

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.

The aerated sequence (b(n))n>=1 = [1, 0, 197, 0, 39005, 0, 7722793, 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 = -200, Q = 1 of the 3-parameter family of divisibility sequences found by Williams and Guy. See A100047 for the connection with Chebyshev polynomials. (End)

Sum_{n >= 1} 1/( a(n) - 1/a(n) ) = 1/196. - Peter Bala, Mar 26 2015

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, 197}, 12] (* Ray Chandler, Aug 11 2015 *)

CROSSREFS

Cf. A097731 for S(n, 198). Row 7 of array A188647. Cf. A000129, A100047.

Sequence in context: A201256 A231462 A188361 * A114050 A268168 A145452

Adjacent sequences:  A097730 A097731 A097732 * A097734 A097735 A097736

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Aug 31 2004

STATUS

approved

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Last modified July 23 22:27 EDT 2017. Contains 289716 sequences.