A001084 a(n) = 20*a(n-1) - a(n-2) with a(0) = 0, a(1) = 3. (Formerly M3167 N1284)

0, 3, 60, 1197, 23880, 476403, 9504180, 189607197, 3782639760, 75463188003, 1505481120300, 30034159217997, 599177703239640, 11953519905574803, 238471220408256420, 4757470888259553597, 94910946544782815520, 1893461460007396756803, 37774318253603152320540

Offset

0,2

Comments

Also 11*x^2+1 is a square. n=11 in PARI script below. - Cino Hilliard, Mar 08 2003

This sequence gives the values of y in solutions of the Diophantine equation x^2 - 11*y^2 = 1; the corresponding x values are in A001085. - Vincenzo Librandi, Nov 12 2010 [edited by Jon E. Schoenfield, May 04 2014]

References

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

"Questions D'Arithmetique", Question 3686, Solution by H.L. Mennessier, Mathesis, 65(4, Supplement) 1956, pp. 1-12.

Links

T. D. Noe, Table of n, a(n) for n = 0..200

H. Brocard, Notes élémentaires sur le problème de Peel, Nouvelle Correspondance Mathématique, 4 (1878), 161-169.

Tanya Khovanova, Recursive Sequences

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

Index entries for linear recurrences with constant coefficients, signature (20,-1).

Formula

Limit_{n->oo} a(n)/a(n-1) = 10 + 3*sqrt(11); for all n in the sequence, 11*n^2 + 1 is a perfect square. - Gregory V. Richardson, Oct 06 2002

a(n) = ((10 + 3*sqrt(11))^n - (10 - 3*sqrt(11))^n) / (2*sqrt(11)). - Gregory V. Richardson, Oct 06 2002

From Mohamed Bouhamida, Sep 20 2006: (Start)

a(n) = 19*(a(n-1) + a(n-2)) - a(n-3).

a(n) = 21*(a(n-1) - a(n-2)) + a(n-3). (End)

G.f.: 3*x/(1 - 20*x + x^2). - G. C. Greubel, Dec 20 2017

E.g.f.: exp(10*x)*sinh(3*sqrt(11)*x)/sqrt(11). - Stefano Spezia, Aug 16 2024

Maple

A001084:=3*z/(1-20*z+z**2); # conjectured by Simon Plouffe in his 1992 dissertation

Mathematica

LinearRecurrence[{20, -1}, {0, 3}, 20] (* T. D. Noe, Dec 19 2011 *)
CoefficientList[Series[3*x/(1 - 20*x + x^2), {x, 0, 50}], x] (* G. C. Greubel, Dec 20 2017 *)
Table[3 ChebyshevU[-1 + n, 10], {n, 0, 18}] (* Herbert Kociemba, Jun 05 2022 *)

Prog

(PARI) nxsqp1(m, n) = { for(x=1, m, y = n*x*x+1; if(issquare(y), print1(x" ")) ) }
(Magma) I:=[0, 3]; [n le 2 select I[n] else 20*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Dec 19 2017
(PARI) x='x+O('x^30); concat([0], Vec(3*x/(1 - 20*x + x^2))) \\ G. C. Greubel, Dec 20 2017

Crossrefs

Equals 3 * A075843.

Cf. A001085, A221762.

Sequence in context: A115490 A065889 A183251 * A137150 A248707 A219870

Adjacent sequences: A001081 A001082 A001083 * A001085 A001086 A001087

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved