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A075839 Numbers k such that 11*k^2 - 2 is a square.

%I A075839 #60 Jul 14 2025 15:19:55
%S A075839 1,19,379,7561,150841,3009259,60034339,1197677521,23893516081,
%T A075839 476672644099,9509559365899,189714514673881,3784780734111721,
%U A075839 75505900167560539,1506333222617099059,30051158552174420641,599516837820871313761,11960285597865251854579
%N A075839 Numbers k such that 11*k^2 - 2 is a square.
%C A075839 Lim_{n -> infinity} a(n)/a(n-1) = 10 + 3*sqrt(11).
%C A075839 Positive values of x (or y) satisfying x^2 - 20xy + y^2 + 18 = 0. - _Colin Barker_, Feb 18 2014
%D A075839 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.
%D A075839 L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine Analysis. AMS Chelsea Publishing, Providence, Rhode Island, 1999, pp. 341-400.
%D A075839 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.
%H A075839 G. C. Greubel, <a href="/A075839/b075839.txt">Table of n, a(n) for n = 1..750</a> (terms 1..200 from Vincenzo Librandi)
%H A075839 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A075839 J. J. O'Connor and E. F. Robertson, <a href="https://mathshistory.st-andrews.ac.uk/HistTopics/Pell/">Pell's Equation</a>
%H A075839 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PellEquation.html">Pell Equation.</a>
%H A075839 <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>
%H A075839 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (20,-1).
%F A075839 11*a(n)^2 - 9*A083043(n)^2 = 2.
%F A075839 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
%F A075839 From _Michael Somos_, Oct 29 2002: (Start)
%F A075839 G.f.: x*(1-x)/(1-20*x+x^2).
%F A075839 a(n) = 20*a(n-1) - a(n-2), n>1. (End)
%F A075839 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
%F A075839 a(-n+1) = a(n). - _Michael Somos_, Apr 18 2003
%F A075839 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
%p A075839 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
%t A075839 LinearRecurrence[{20,-1},{1,19},20] (* _Harvey P. Dale_, Apr 13 2012 *)
%t A075839 Rest@CoefficientList[Series[x*(1-x)/(1-20x+x^2), {x, 0, 20}], x] (* _Vincenzo Librandi_, Feb 20 2014 *)
%t A075839 a[c_, n_] := Module[{},
%t A075839    p := Length[ContinuedFraction[ Sqrt[ c]][[2]]];
%t A075839    d := Denominator[Convergents[Sqrt[c], n p]];
%t A075839    t := Table[d[[1 + i]], {i, 0, Length[d] - 1, p}];
%t A075839    Return[t];
%t A075839 ] (* Complement of A041015 *)
%t A075839 a[11, 20] (* _Gerry Martens_, Jun 07 2015 *)
%o A075839 (PARI) a(n)=subst(poltchebi(n+1)+poltchebi(n),x,10)/11
%o A075839 (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
%o A075839 (Sage)
%o A075839 def A075839_list(prec):
%o A075839     P.<x> = PowerSeriesRing(ZZ, prec)
%o A075839     return P( x*(1-x)/(1-20*x+x^2) ).list()
%o A075839 a=A075839_list(20); a[1:] # _G. C. Greubel_, Dec 06 2019
%o A075839 (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
%Y A075839 Row 20 of array A094954.
%Y A075839 Cf. A075844, A221762, A041015.
%Y A075839 Cf. similar sequences listed in A238379.
%K A075839 easy,nonn
%O A075839 1,2
%A A075839 _Gregory V. Richardson_, Oct 14 2002
%E A075839 More terms from _Colin Barker_, Feb 18 2014
%E A075839 Offset changed to 1 by _G. C. Greubel_, Dec 06 2019