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 A075839 Numbers k such that 11*k^2 - 2 is a square. 10
 1, 19, 379, 7561, 150841, 3009259, 60034339, 1197677521, 23893516081, 476672644099, 9509559365899, 189714514673881, 3784780734111721, 75505900167560539, 1506333222617099059, 30051158552174420641, 599516837820871313761, 11960285597865251854579 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Lim_{n -> infinity} a(n)/a(n-1) = 10 + 3*sqrt(11). Positive values of x (or y) satisfying x^2 - 20xy + y^2 + 18 = 0. - Colin Barker, Feb 18 2014 REFERENCES A. H. Beiler, "The Pellian", ch. 22 in Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. Dover, New York, New York, pp. 248-268, 1966. L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine Analysis. AMS Chelsea Publishing, Providence, Rhode Island, 1999, pp. 341-400. Peter G. L. Dirichlet, Lectures on Number Theory (History of Mathematics Source Series, V. 16); American Mathematical Society, Providence, Rhode Island, 1999, pp. 139-147. LINKS G. C. Greubel, Table of n, a(n) for n = 1..750 (terms 1..200 from Vincenzo Librandi) Tanya Khovanova, Recursive Sequences J. J. O'Connor and E. F. Robertson, Pell's Equation Eric Weisstein's World of Mathematics, Pell Equation. Index entries for two-way infinite sequences Index entries for linear recurrences with constant coefficients, signature (20,-1). FORMULA 11*a(n)^2 - 9*A083043(n)^2 = 2. a(n) = ((3+sqrt(11))*(10+3*sqrt(11))^(n-1) - (3-sqrt(11))*(10-3*sqrt(11))^(n-1) )/(2*sqrt(11)). - Dean Hickerson, Dec 09 2002 From Michael Somos, Oct 29 2002: (Start) G.f.: x*(1-x)/(1-20*x+x^2). a(n) = 20*a(n-1) - a(n-2), n>1. (End) Let q(n, x) = Sum_{i=0..n} x^(n-i)*binomial(2*n-i, i) then a(n) = q(n, 18). - Benoit Cloitre, Dec 06 2002 a(-n+1) = a(n). - Michael Somos, Apr 18 2003 E.g.f.: (1/11)*exp(10*x)*(11*cosh(3*sqrt(11)*x) - 3*sqrt(11)*sinh(3*sqrt(11)*x)) - 1. - Stefano Spezia, Dec 06 2019 MAPLE seq(coeff(series( x*(1-x)/(1-20*x+x^2), x, n+1), x, n), n = 1..20); # G. C. Greubel, Dec 06 2019 MATHEMATICA LinearRecurrence[{20, -1}, {1, 19}, 20] (* Harvey P. Dale, Apr 13 2012 *) Rest@CoefficientList[Series[x*(1-x)/(1-20x+x^2), {x, 0, 20}], x] (* Vincenzo Librandi, Feb 20 2014 *) a[c_, n_] := Module[{}, p := Length[ContinuedFraction[ Sqrt[ c]][[2]]]; d := Denominator[Convergents[Sqrt[c], n p]]; t := Table[d[[1 + i]], {i, 0, Length[d] - 1, p}]; Return[t]; ] (* Complement of A041015 *) a[11, 20] (* Gerry Martens, Jun 07 2015 *) PROG (PARI) a(n)=subst(poltchebi(n+1)+poltchebi(n), x, 10)/11 (Magma) I:=[1, 19]; [n le 2 select I[n] else 20*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Feb 20 2014 (Sage) def A075839_list(prec): P. = PowerSeriesRing(ZZ, prec) return P( x*(1-x)/(1-20*x+x^2) ).list() a=A075839_list(20); a[1:] # G. C. Greubel, Dec 06 2019 (GAP) a:=[1, 19];; for n in [3..20] do a[n]:=20*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Dec 06 2019 CROSSREFS Row 20 of array A094954. Cf. A075844, A221762, A041015. Cf. similar sequences listed in A238379. Sequence in context: A041686 A263371 A023283 * A158592 A072359 A222835 Adjacent sequences: A075836 A075837 A075838 * A075840 A075841 A075842 KEYWORD easy,nonn AUTHOR Gregory V. Richardson, Oct 14 2002 EXTENSIONS More terms from Colin Barker, Feb 18 2014 Offset changed to 1 by G. C. Greubel, Dec 06 2019 STATUS approved

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