|
%I #30 Jan 26 2020 17:45:01
%S 0,3,33,360,3927,42837,467280,5097243,55602393,606529080,6616217487,
%T 72171863277,787274278560,8587845200883,93679022931153,
%U 1021881407041800,11147016454528647,121595299592773317
%N Numbers n such that 13*n^2 + 4 is a square.
%C Lim_{n->infinity} a(n)/a(n-1) = (11 + sqrt(13))/2.
%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.
%D S. Falcon, Relationships between Some k-Fibonacci Sequences, Applied Mathematics, 2014, 5, 2226-2234; http://www.scirp.org/journal/am; http://dx.doi.org/10.4236/am.2014.515216
%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="http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Pell.html">Pell's Equation</a>
%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 (11,-1).
%F a(n) = ((11 + 3*sqrt(13))^n - (11 - 3*sqrt(13))^n) / ((2^n) * sqrt(13)).
%F From _Philippe Deléham_, Nov 17 2008: (Start)
%F a(n) = 11*a(n-1) - a(n-2) with a(1)=0 and a(2)=3.
%F G.f.: 3x^2/(1-11x+x^2). (End)
%F a(n) = A006190(2*n). - _Vladimir Reshetnikov_, Sep 16 2016
%t LinearRecurrence[{11,-1},{0,3},20] (* _Harvey P. Dale_, Dec 27 2011 *)
%t Table[Fibonacci[2n, 3], {n, 0, 20}] (* _Vladimir Reshetnikov_, Sep 16 2016 *)
%Y Cf. A006190.
%K nonn,easy
%O 1,2
%A _Gregory V. Richardson_, Oct 14 2002
|
