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A075869 Numbers k such that 5*k^2 - 9 is a square. 0

%I #31 May 15 2019 08:43:35

%S 3,51,915,16419,294627,5286867,94868979,1702354755,30547516611,

%T 548152944243,9836205479763,176503545691491,3167227616967075,

%U 56833593559715859,1019837456457918387,18300240622682815107

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

%C Lim. n-> Inf. a(n)/a(n-1) = phi^6 = 9 + 4*sqrt(5).

%D 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 L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine Analysis. AMS Chelsea Publishing, Providence, Rhode Island, 1999, pp. 341-400.

%D 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 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H J. J. O'Connor and E. F. Robertson, <a href="https://web.archive.org/web/20170729132724/http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Pell.html">Pell's Equation</a> [From the Internet Archive Wayback machine]

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PellEquation.html">Pell Equation.</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (18,-1).

%F 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 G.f.: 3*x*(1-x)/(1-18*x+x^2). [_Philippe Deléham_, Nov 17 2008; corrected by _Georg Fischer_, May 15 2019]

%t LinearRecurrence[{18, -1}, {3, 51}, 20] (* _Harvey P. Dale_, Dec 27 2018 *)

%Y Cf. 3*A007805.

%K nonn,easy

%O 1,1

%A _Gregory V. Richardson_, Oct 16 2002

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Last modified March 29 06:57 EDT 2024. Contains 371265 sequences. (Running on oeis4.)