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A075869 Numbers k such that 5*k^2 - 9 is a square.

%I A075869 #35 Jul 23 2025 14:43:44
%S A075869 3,51,915,16419,294627,5286867,94868979,1702354755,30547516611,
%T A075869 548152944243,9836205479763,176503545691491,3167227616967075,
%U A075869 56833593559715859,1019837456457918387,18300240622682815107
%N A075869 Numbers k such that 5*k^2 - 9 is a square.
%C A075869 Lim. n-> Inf. a(n)/a(n-1) = phi^6 = 9 + 4*sqrt(5).
%D A075869 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 A075869 L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine Analysis. AMS Chelsea Publishing, Providence, Rhode Island, 1999, pp. 341-400.
%D A075869 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 A075869 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A075869 J. J. O'Connor and E. F. Robertson, <a href="https://mathshistory.st-andrews.ac.uk/HistTopics/Pell/">Pell's Equation</a>
%H A075869 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PellEquation.html">Pell Equation.</a>
%H A075869 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (18,-1).
%F A075869 a(n) = 3*sqrt(5)/10*((2+sqrt(5))^(2*n-1)-(2-sqrt(5))^(2*n-1)) = 18*a(n-1) - a(n-2).
%F A075869 G.f.: 3*x*(1-x)/(1-18*x+x^2). [_Philippe Deléham_, Nov 17 2008; corrected by _Georg Fischer_, May 15 2019]
%t A075869 LinearRecurrence[{18, -1}, {3, 51}, 20] (* _Harvey P. Dale_, Dec 27 2018 *)
%Y A075869 Cf. 3*A007805.
%K A075869 nonn,easy
%O A075869 1,1
%A A075869 _Gregory V. Richardson_, Oct 16 2002