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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 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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