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%I #21 Sep 08 2022 08:45:07
%S 3,22,173,1362,10723,84422,664653,5232802,41197763,324349302,
%T 2553596653,20104423922,158281794723,1246149933862,9810917676173,
%U 77241191475522,608118614128003,4787707721548502,37693543158260013,296760637544531602,2336391557197992803
%N Bisection (odd part) of Chebyshev sequence with Diophantine property.
%C 3*a(n)^2 - 5*b(n)^2 = 7, with the companion sequence b(n)= A077243(n).
%C The even part is A077246(n) with Diophantine companion A077245(n).
%H Colin Barker, <a href="/A077244/b077244.txt">Table of n, a(n) for n = 0..1000</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,-1).
%F a(n)= (2*T(n+1, 4)+T(n, 4))/3, with T(n, x) Chebyshev's polynomials of the first kind, A053120. T(n, 4)= A001091(n).
%F G.f.: (3-2*x)/(1-8*x+x^2).
%F From _Colin Barker_, Oct 12 2015: (Start)
%F a(n) = (((4-sqrt(15))^n * (-10+3*sqrt(15)) + (4+sqrt(15))^n * (10+3*sqrt(15)))) / (2*sqrt(15)).
%F a(n) = 8*a(n-1) - a(n-2).
%F (End)
%e 22 = a(1) = sqrt((5*A077243(1)^2 + 7)/3) = sqrt((5*17^2 + 7)/3) = sqrt(484) = 22.
%t LinearRecurrence[{8, -1}, {3, 22}, 25] (* _Vincenzo Librandi_, Oct 12 2015 *)
%o (PARI) Vec((3-2*x)/(1-8*x+x^2) + O(x^40)) \\ _Colin Barker_, Oct 12 2015
%o (Magma) I:=[3,22]; [n le 2 select I[n] else 8*Self(n-1)-Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Oct 12 2015
%K nonn,easy
%O 0,1
%A _Wolfdieter Lang_, Nov 08 2002
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