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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
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
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
FORMULA
Limit_{n->infinity} 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
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.
Sequence in context: A115490 A065889 A183251 * A137150 A248707 A219870
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified March 18 22:34 EDT 2024. Contains 370951 sequences. (Running on oeis4.)