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%I #54 Dec 31 2023 11:27:58
%S 0,6,228,8658,328776,12484830,474094764,18003116202,683644320912,
%T 25960481078454,985814636660340,37434995712014466,1421544022419889368,
%U 53981237856243781518,2049865494514843808316,77840907553707820934490,2955904621546382351702304
%N a(n) = 38*a(n-1) - a(n-2), with a(0)=0, a(1)=6.
%C This sequence gives the values of y in solutions of the Diophantine equation x^2 - 10*y^2 = 1. The corresponding x values are in A078986. - _Vincenzo Librandi_, Aug 08 2010 [edited by _Jon E. Schoenfield_, May 04 2014]
%H Indranil Ghosh, <a href="/A084070/b084070.txt">Table of n, a(n) for n = 0..632</a>
%H Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Nemeth/nemeth7.html">Ellipse Chains and Associated Sequences</a>, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
%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 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (38,-1).
%F Numbers k such that 10*k^2 = floor(k*sqrt(10)*ceiling(k*sqrt(10))).
%F From _Mohamed Bouhamida_, Sep 20 2006: (Start)
%F a(n) = 37*(a(n-1) + a(n-2)) - a(n-3).
%F a(n) = 39*(a(n-1) - a(n-2)) + a(n-3). (End)
%F From _R. J. Mathar_, Feb 19 2008: (Start)
%F O.g.f.: 6*x/(1 - 38*x + x^2).
%F a(n) = 6*A078987(n-1). (End)
%F a(n) = 6*ChebyshevU(n-1, 19). - _G. C. Greubel_, Jan 12 2020
%F a(n) = A005668(2*n). - _Michael Somos_, Feb 24 2023
%e G.f. = 6*x + 228*x^2 + 8658*x^3 + 328776*x^4 + ... - _Michael Somos_, Feb 24 2023
%p seq( simplify(6*ChebyshevU(n-1, 19)), n=0..20); # _G. C. Greubel_, Jan 12 2020
%t LinearRecurrence[{38,-1},{0,6},30] (* _Harvey P. Dale_, Nov 01 2011 *)
%t 6*ChebyshevU[Range[20]-2, 19] (* _G. C. Greubel_, Jan 12 2020 *)
%o (PARI) u=0; v=6; for(n=2,20, w=38*v-u; u=v; v=w; print1(w,","))
%o (PARI) vector(21, n, 6*polchebyshev(n-2, 2, 19) ) \\ _G. C. Greubel_, Jan 12 2020
%o (Magma) I:=[0,6]; [n le 2 select I[n] else 38*Self(n-1) - Self(n-2): n in [1..20]]; // _G. C. Greubel_, Jan 12 2020
%o (Sage) [6*chebyshev_U(n-1, 19) for n in (0..20)] # _G. C. Greubel_, Jan 12 2020
%o (GAP) a:=[0,6];; for n in [3..20] do a[n]:=38*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, Jan 12 2020
%Y Cf. A001078, A001109, A001353, A001653, A005668, A060645, A084068, A084069, A221874.
%Y Cf. A078986. - _Vincenzo Librandi_, Apr 14 2010
%K nonn
%O 0,2
%A _Benoit Cloitre_, May 10 2003
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