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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. 4
 1, 199, 39401, 7801199, 1544598001, 305822602999, 60551330795801, 11988857674965599, 2373733268312392801, 469987198268178808999, 93055091523831091789001, 18424438134520287995413199, 3647945695543493192000024401, 722274823279477131728009418199 (list; graph; refs; listen; history; text; internal format)
 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

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