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%I #25 Sep 08 2022 08:45:01
%S 1,7,34,163,781,3742,17929,85903,411586,1972027,9448549,45270718,
%T 216905041,1039254487,4979367394,23857582483,114308545021,
%U 547685142622,2624117168089,12572900697823,60240386321026,288629030907307,1382904768215509,6625894810170238,31746569282635681,152106951603008167
%N a(n) = 5*a(n-1) - a(n-2) with a(0)=1, a(1)=7.
%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 122-125, 194-196.
%H G. C. Greubel, <a href="/A055271/b055271.txt">Table of n, a(n) for n = 0..1000</a>
%H I. Adler, <a href="http://www.fq.math.ca/Scanned/7-2/adler.pdf">Three Diophantine equations - Part II</a>, Fib. Quart., 7 (1969), pp. 181-193.
%H E. I. Emerson, <a href="http://www.fq.math.ca/Scanned/7-3/emerson.pdf">Recurrent Sequences in the Equation DQ^2=R^2+N</a>, Fib. Quart., 7 (1969), pp. 231-242.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-1).
%F a(n) = (7*(((5+sqrt(21))/2)^n - ((5-sqrt(21))/2)^n) - (((5+sqrt(21))/2)^(n-1) - ((5-sqrt(21))/2)^(n-1)))/sqrt(21).
%F G.f.: (1+2*x)/(1-5*x+x^2).
%F a(n) = (-1)^n*Sum_{k = 0..n} A238731(n,k)*(-8)^k. - _Philippe Deléham_, Mar 05 2014
%F a(n) = ChebyshevT(n, 5/2) + (9/2)*ChebyshevU(n-1,5/2) = ChebyshevU(n, 5/2) + 2*ChebyshevU(n-1, 5/2). - _G. C. Greubel_, Mar 16 2020
%p A055271:= n-> simplify(ChebyshevU(n, 5/2) + 2*ChebyshevU(n-1, 5/2)); seq(A055271(n), n=0..30); # _G. C. Greubel_, Mar 16 2020
%t LinearRecurrence[{5,-1}, {1,7}, 30] (* _G. C. Greubel_, Mar 16 2020 *)
%o (Magma) I:=[1,7]; [n le 2 select I[n] else 5*Self(n-1) - Self(n-2): n in [1..30]]; // _G. C. Greubel_, Mar 16 2020
%o (Sage) [chebyshev_U(n, 5/2) + 2*chebyshev_U(n-1, 5/2) for n in (0..30)] # _G. C. Greubel_, Mar 16 2020
%Y Cf. A030221.
%K easy,nonn
%O 0,2
%A _Barry E. Williams_, May 10 2000
%E Terms a(22) onward added by _G. C. Greubel_, Mar 16 2020
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