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%I #24 Dec 07 2019 12:18:24
%S 2,25,623,15550,388127,9687625,241802498,6035374825,150642568127,
%T 3760028828350,93850078140623,2342491924687225,58468448039040002,
%U 1459368709051312825,36425749278243780623,909184363247043202750
%N a(n) = 25*a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 25.
%C A Chebyshev T-sequence with Diophantine property.
%C a(n) gives the general (nonnegative integer) solution of the Pell equation a^2 - 69*(3*b)^2 =+4 together with the companion sequence b(n)=A097780(n-1), n>=0.
%D O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).
%H Indranil Ghosh, <a href="/A090733/b090733.txt">Table of n, a(n) for n = 0..714</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H <a href="/index/Rea#recur1">Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)</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 (25,-1)
%F a(n) = S(n, 25) - S(n-2, 25) = 2*T(n, 25/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 25)=A097780(n). U-, resp. T-, are Chebyshev's polynomials of the second, resp. first, kind. See A049310 and A053120.
%F a(n) = ap^n + am^n, with ap := (25+3*sqrt(69))/2 and am := (25-3*sqrt(69))/2.
%F G.f.: (2-25*x)/(1-25*x+x^2).
%e (x,y) =(2,0), (25;1), (623;25), (15550;624), ... give the nonnegative integer solutions to x^2 - 69*(3*y)^2 =+4.
%t a[0] = 2; a[1] = 25; a[n_] := 25a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 15}] (* _Robert G. Wilson v_, Jan 30 2004 *)
%o (Sage) [lucas_number2(n,25,1) for n in range(0,20)] # _Zerinvary Lajos_, Jun 26 2008
%Y Cf. A046069, A082974.
%Y a(n)=sqrt(4 + 69*(3*A097780(n-1))^2), n>=1.
%Y Cf. A077428, A078355 (Pell +4 equations).
%Y Cf. A097779 for 2*T(n, 23/2).
%K easy,nonn
%O 0,1
%A Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 18 2004
%E Extension, Chebyshev and Pell comments from _Wolfdieter Lang_, Aug 31 2004
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