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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 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

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Last modified September 15 16:35 EDT 2024. Contains 375938 sequences. (Running on oeis4.)