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%I #22 Dec 24 2021 15:34:37
%S 2,19,359,6802,128879,2441899,46267202,876634939,16609796639,
%T 314709501202,5962870726199,112979834296579,2140653980908802,
%U 40559445802970659,768488816275533719,14560728063432170002
%N A Chebyshev T-sequence with Diophantine property.
%C a(n) gives the general (positive integer) solution of the Pell equation a^2 - 357*b^2 =+4 with companion sequence b(n)=A078368(n-1), n>=1.
%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 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 (19,-1).
%F a(n)=19*a(n-1)-a(n-2), n >= 1; a(-1)=19, a(0)=2.
%F a(n) = S(n, 19) - S(n-2, 19) = 2*T(n, 19/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 19)=A078368(n). U-, resp. T-, are Chebyshev's polynomials of the second, resp. first, case. See A049310 and A053120.
%F G.f.: (2-19*x)/(1-19*x+x^2).
%F a(n) = ap^n + am^n, with ap := (19+sqrt(357))/2 and am := (19-sqrt(357))/2.
%t a[0] = 2; a[1] = 19; a[n_] := 19a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 15}] (* _Robert G. Wilson v_, Jan 30 2004 *)
%t LinearRecurrence[{19,-1},{2,19},20] (* _Harvey P. Dale_, Dec 24 2021 *)
%o (Sage) [lucas_number2(n,19,1) for n in range(0,20)] # _Zerinvary Lajos_, Jun 27 2008
%Y a(n)=sqrt(4 + 357*A078368(n-1)^2), n>=1, (Pell equation d=357, +4).
%Y Cf. A077428, A078355 (Pell +4 equations).
%K nonn,easy
%O 0,1
%A _Wolfdieter Lang_, Nov 29 2002
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