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 A097730 Pell equation solutions (6*b(n))^2 - 37*a(n)^2 = -1 with b(n)=A097729(n), n>=0. 5
 1, 145, 21169, 3090529, 451196065, 65871534961, 9616792908241, 1403985893068225, 204972323595052609, 29924555258984612689, 4368780095488158399985, 637811969386012141785121, 93116178750262284542227681, 13594324285568907531023456305 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Indranil Ghosh, Table of n, a(n) for n = 0..461 Tanya Khovanova, Recursive Sequences Giovanni Lucca, Integer Sequences and Circle Chains Inside a Hyperbola, Forum Geometricorum (2019) Vol. 19, 11-16. Index entries for linear recurrences with constant coefficients, signature (146, -1). FORMULA a(n) = S(n, 2*73) - S(n-1, 2*73) = T(2*n+1, sqrt(37)/sqrt(37), 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, 12*I) with the imaginary unit I and Chebyshev polynomials S(n, x) with coefficients shown in A049310. G.f.: (1-x)/(1-146*x+x^2). a(n) = 146*a(n-1) - a(n-2), n>1 ; a(0)=1, a(1)=145 . - Philippe Deléham, Nov 18 2008 EXAMPLE (x,y) = (6,1), (882,145), (128766,21169), ... give the positive integer solutions to x^2 - 37*y^2 =-1. MATHEMATICA LinearRecurrence[{146, -1}, {1, 145}, 12] (* Ray Chandler, Aug 12 2015 *) PROG (PARI) my(x='x+O('x^20)); Vec((1-x)/(1-146*x+x^2)) \\ G. C. Greubel, Aug 01 2019 (MAGMA) I:=[1, 145]; [n le 2 select I[n] else 146*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Aug 01 2019 (Sage) ((1-x)/(1-146*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019 (GAP) a:=[1, 145];; for n in [3..20] do a[n]:=146*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019 CROSSREFS Cf. A097729 for S(n, 146). Row 6 of array A188647. Sequence in context: A012813 A031612 A226849 * A283520 A265439 A060720 Adjacent sequences:  A097727 A097728 A097729 * A097731 A097732 A097733 KEYWORD nonn,easy AUTHOR Wolfdieter Lang, Aug 31 2004 STATUS approved

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Last modified October 19 11:09 EDT 2019. Contains 328216 sequences. (Running on oeis4.)