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A001084 a(n) = 20*a(n-1) - a(n-2) with a(0) = 0, a(1) = 3.
(Formerly M3167 N1284)
5
0, 3, 60, 1197, 23880, 476403, 9504180, 189607197, 3782639760, 75463188003, 1505481120300, 30034159217997, 599177703239640, 11953519905574803, 238471220408256420, 4757470888259553597, 94910946544782815520, 1893461460007396756803, 37774318253603152320540 (list; graph; refs; listen; history; text; internal format)
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

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

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

Tanya Khovanova, Recursive Sequences

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

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

FORMULA

Lim 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

a(n) = 19*(a(n-1)+a(n-2))-a(n-3). a(n) = 21*(a(n-1)-a(n-2))+a(n-3). - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 20 2006

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 *)

PROG

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

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

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Last modified March 23 20:21 EDT 2017. Contains 283957 sequences.