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%I #33 Sep 08 2022 08:45:07
%S 0,6,120,2394,47760,952806,19008360,379214394,7565279520,150926376006,
%T 3010962240600,60068318435994,1198355406479280,23907039811149606,
%U 476942440816512840,9514941776519107194,189821893089565631040
%N Numbers n such that 11*n^2 + 4 is a square.
%C Lim. n-> Inf. a(n)/a(n-1) = 10 + 3*sqrt(11).
%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 Harvey P. Dale, <a href="/A075844/b075844.txt">Table of n, a(n) for n = 0..750</a>
%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 (20,-1).
%F a(n) = ((10+3*sqrt(11))^n - (10-3*sqrt(11))^n) / sqrt(11).
%F a(n) = 20*a(n-1) - a(n-2).
%F G.f.: 6*x/(1 - 20*x + x^2).
%F a(n) = (1/3)*(A075839(n+1) - A075839(n)), n>=1. - _N. J. A. Sloane_, Sep 22 2004
%F a(n) = 6*A075843(n). - _R. J. Mathar_, Jul 03 2011
%p seq(coeff(series(6*x/(1-20*x+x^2), x, n+1), x, n), n = 0..20); # _G. C. Greubel_, Dec 06 2019
%t LinearRecurrence[{20,-1},{0,6},20] (* _Harvey P. Dale_, May 28 2012 *)
%o (PARI) my(x='x+O('x^20)); concat([0], Vec(6*x/(1-20*x+x^2))) \\ _G. C. Greubel_, Dec 06 2019
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 20); [0] cat Coefficients(R!( 6*x/(1 - 20*x + x^2) )); // _G. C. Greubel_, Dec 06 2019
%o (Sage)
%o def A075844_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( 6*x/(1-20*x+x^2) ).list()
%o A075844_list(20) # _G. C. Greubel_, Dec 06 2019
%o (GAP) a:=[0,6];; for n in [3..20] do a[n]:=20*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, Dec 06 2019
%Y Cf. A221762.
%K nonn,easy
%O 0,2
%A _Gregory V. Richardson_, Oct 14 2002
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