A084070 a(n) = 38*a(n-1) - a(n-2), with a(0)=0, a(1)=6.

0, 6, 228, 8658, 328776, 12484830, 474094764, 18003116202, 683644320912, 25960481078454, 985814636660340, 37434995712014466, 1421544022419889368, 53981237856243781518, 2049865494514843808316, 77840907553707820934490, 2955904621546382351702304

Offset: 0

Author

Benoit Cloitre, May 10 2003

Keywords

• nonn

Comments

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]

Examples

			G.f. = 6*x + 228*x^2 + 8658*x^3 + 328776*x^4 + ... - _Michael Somos_, Feb 24 2023
		

Links

• 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.
• A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929. See Vol. 1, page xxxv.
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (38,-1).

Crossrefs

Cf. A001078, A001109, A001353, A001653, A005668, A060645, A084068, A084069, A221874.

Cf. A078986. - Vincenzo Librandi, Apr 14 2010

Programs

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

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

Maple

seq( simplify(6*ChebyshevU(n-1, 19)), n=0..20); # G. C. Greubel, Jan 12 2020

Mathematica

LinearRecurrence[{38,-1},{0,6},30] (* Harvey P. Dale, Nov 01 2011 *)
6*ChebyshevU[Range[20]-2, 19] (* G. C. Greubel, Jan 12 2020 *)

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

Sage

[6*chebyshev_U(n-1, 19) for n in (0..20)] # G. C. Greubel, Jan 12 2020

Formula

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) = 6*ChebyshevU(n-1, 19). - G. C. Greubel, Jan 12 2020

a(n) = A005668(2*n). - Michael Somos, Feb 24 2023