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%I #21 Jan 01 2024 11:05:43
%S 2,13,102,803,6322,49773,391862,3085123,24289122,191227853,1505533702,
%T 11853041763,93318800402,734697361453,5784260091222,45539383368323,
%U 358530806855362,2822707071474573,22223125764941222
%N Bisection (even part) of Chebyshev sequence with Diophantine property.
%C 3*a(n)^2 - 5*b(n)^2 = 7, with the companion sequence b(n)= A077245(n).
%C The odd part is A077244(n) with Diophantine companion A077243(n).
%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)= 8*a(n-1) - a(n-2), a(-1) := 3, a(0)=2.
%F a(n)= (T(n+1, 4)+2*T(n, 4))/3, with T(n, x) Chebyshev's polynomials of the first kind, A053120. T(n, 4)= A001091(n).
%F G.f.: (2-3*x)/(1-8*x+x^2).
%e 13 = a(1) = sqrt((5*A077245(1)^2 + 7)/3) = sqrt((5*10^2 + 7)/3) = sqrt(169) = 13.
%t LinearRecurrence[{8,-1},{2,13},30] (* _Harvey P. Dale_, Apr 30 2012 *)
%K nonn,easy
%O 0,1
%A _Wolfdieter Lang_, Nov 08 2002
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