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

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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).

Crossrefs

Equals 3 * A075843.

Cf. A001085, A221762.

Programs

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

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 *)

PARI

nxsqp1(m,n) = { for(x=1,m, y = n*x*x+1; if(issquare(y),print1(x" ")) ) }

PARI

x='x+O('x^30); concat([0], Vec(3*x/(1 - 20*x + x^2))) \\ G. C. Greubel, Dec 20 2017

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