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%I #54 Feb 25 2023 15:18:56
%S 1,37,1405,53353,2026009,76934989,2921503573,110940200785,
%T 4212806126257,159975692596981,6074863512559021,230684837784645817,
%U 8759948972303982025,332647376109766671133,12631840343198829521029,479677285665445755127969,18215105014943739865341793,691694313282196669127860165
%N Pell equation solutions (3*b(n))^2 - 10*a(n)^2 = -1 with b(n) = A097314(n), n >= 0.
%C Hypotenuses of primitive Pythagorean triples in A195616 and A195617. - _Clark Kimberling_, Sep 22 2011
%H Indranil Ghosh, <a href="/A097315/b097315.txt">Table of n, a(n) for n = 0..631</a>
%H A. J. C. Cunningham, <a href="https://archive.org/details/binomialfactoris01cunn/page/n46/mode/1up">Binomial Factorisations</a>, Vols. 1-9, Hodgson, London, 1923-1929. See Vol. 1, page xxxv.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H Giovanni Lucca, <a href="http://forumgeom.fau.edu/FG2019volume19/FG201902index.html">Integer Sequences and Circle Chains Inside a Hyperbola</a>, Forum Geometricorum (2019) Vol. 19, 11-16.
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (38,-1).
%F a(n) = S(n, 38) - S(n-1, 38) = T(2*n+1, sqrt(10))/sqrt(10), with Chebyshev polynomials of the second 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.
%F a(n) = ((-1)^n)*S(2*n, 6*i) with the imaginary unit i and Chebyshev polynomials S(n, x) with coefficients shown in A049310.
%F G.f.: (1-x)/(1-38*x+x^2).
%F a(n) = 38*a(n-1) - a(n-2) for n > 1. - _Philippe Deléham_, Nov 18 2008
%F a(n) = sqrt(2+(19-6*sqrt(10))^(1+2*n) + (19+6*sqrt(10))^(1+2*n))/(2*sqrt(10)). - _Gerry Martens_, Jun 04 2015
%F a(n) = A078987(n) - A078987(n-1). - _R. J. Mathar_, Dec 05 2015
%F a(n) = A005668(2*n+1). - _Michael Somos_, Feb 24 2023
%e (x,y) = (3,1), (117,37), (4443,1405), ... give the positive integer solutions to x^2 - 10*y^2 = -1.
%e G.f. = 1 + 37*x + 1405*x^2 + 53353*x^3 + ... - _Michael Somos_, Feb 24 2023
%t CoefficientList[Series[(1-x)/(1-38x+x^2), {x,0,20}], x] (* _Michael De Vlieger_, Feb 04 2017 *)
%t LinearRecurrence[{38,-1}, {1,37}, 21] (* _G. C. Greubel_, Aug 01 2019 *)
%o (PARI) Vec((1-x)/(1-38*x+x^2) + O(x^20)) \\ _Michel Marcus_, Jun 04 2015
%o (Magma) I:=[1, 37]; [n le 2 select I[n] else 38*Self(n-1) - Self(n-2): n in [1..30]]; // _G. C. Greubel_, Aug 01 2019
%o (Sage) ((1-x)/(1-38*x+x^2)).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, Aug 01 2019
%o (GAP) a:=[1,37];; for n in [3..20] do a[n]:=38*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, Aug 01 2019
%Y Cf. A078987 (partial sums), A097314, A195616, A195617, A049310, A053120, A005668.
%Y Row 3 of array A188647.
%Y Cf. A221874.
%Y Cf. similar sequences listed in A238379.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Aug 31 2004
%E Typo in recurrence formula corrected by Laurent Bonaventure (bonave(AT)free.fr), Oct 03 2010
%E More terms added by _Indranil Ghosh_, Feb 04 2017