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A084070
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a(n) = 38*a(n-1) - a(n-2), with a(0)=0, a(1)=6.
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5
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0, 6, 228, 8658, 328776, 12484830, 474094764, 18003116202, 683644320912, 25960481078454, 985814636660340, 37434995712014466, 1421544022419889368, 53981237856243781518, 2049865494514843808316, 77840907553707820934490, 2955904621546382351702304
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OFFSET
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0,2
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COMMENTS
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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]
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LINKS
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Indranil Ghosh, Table of n, a(n) for n = 0..632
Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (38,-1).
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FORMULA
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Numbers k such that 10*k^2 = floor(k*sqrt(10)*ceiling(k*sqrt(10))).
From Mohamed Bouhamida, Sep 20 2006: (Start)
a(n) = 37*(a(n-1) + a(n-2)) - a(n-3).
a(n) = 39*(a(n-1) - a(n-2)) + a(n-3). (End)
From R. J. Mathar, Feb 19 2008: (Start)
O.g.f.: 6*x/(1 - 38*x + x^2).
a(n) = 6*A078987(n-1). (End)
a(n) = (sqrt(10)/20)*( (19 + 6*sqrt(10))^n - (19 - 6*sqrt(10))^n ), with n >= 0. - Paolo P. Lava, Jul 11 2008
a(n) = 6*ChebyshevU(n-1, 19). - G. C. Greubel, Jan 12 2020
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MAPLE
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seq( simplify(6*ChebyshevU(n-1, 19)), n=0..20); # G. C. Greubel, Jan 12 2020
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MATHEMATICA
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LinearRecurrence[{38, -1}, {0, 6}, 30] (* Harvey P. Dale, Nov 01 2011 *)
6*ChebyshevU[Range[20]-2, 19] (* G. C. Greubel, Jan 12 2020 *)
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PROG
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(PARI) u=0; v=6; for(n=2, 20, w=38*v-u; u=v; v=w; print1(w, ", "))
(PARI) vector(21, n, 6*polchebyshev(n-2, 2, 19) ) \\ G. C. Greubel, Jan 12 2020
(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
(Sage) [6*chebyshev_U(n-1, 19) for n in (0..20)] # G. C. Greubel, Jan 12 2020
(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
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CROSSREFS
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Cf. A001078, A001109, A001353, A001653, A060645, A084068, A084069, A221874.
Cf. A078986. - Vincenzo Librandi, Apr 14 2010
Sequence in context: A166502 A173083 A338297 * A282736 A277293 A177043
Adjacent sequences: A084067 A084068 A084069 * A084071 A084072 A084073
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KEYWORD
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nonn
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AUTHOR
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Benoit Cloitre, May 10 2003
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STATUS
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approved
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