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A078365 A Chebyshev T-sequence with Diophantine property.

%I A078365 #28 Jan 01 2024 11:05:47
%S A078365 2,15,223,3330,49727,742575,11088898,165590895,2472774527,36926027010,
%T A078365 551417630623,8234338432335,122963658854402,1836220544383695,
%U A078365 27420344506901023,409468947059131650
%N A078365 A Chebyshev T-sequence with Diophantine property.
%C A078365 a(n) gives the general (positive integer) solution of the Pell equation a^2 - 221*b^2 =+4 with companion sequence b(n)=A078364(n-1), n>=1.
%D A078365 O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).
%H A078365 Harvey P. Dale, <a href="/A078365/b078365.txt">Table of n, a(n) for n = 0..851</a>
%H A078365 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A078365 <a href="/index/Rea#recur1">Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)</a>
%H A078365 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A078365 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (15,-1).
%F A078365 a(n)=15*a(n-1)-a(n-2), n >= 1; a(-1)=15, a(0)=2.
%F A078365 a(n) = S(n, 15) - S(n-2, 15) = 2*T(n, 15/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 15)=A078364(n). U-, resp. T-, are Chebyshev's polynomials of the second, resp. first, case. See A049310 and A053120.
%F A078365 G.f.: (2-15*x)/(1-15*x+x^2).
%F A078365 a(n) = ap^n + am^n, with ap := (15+sqrt(221))/2 and am := (15-sqrt(221))/2.
%t A078365 a[0] = 2; a[1] = 15; a[n_] := 15a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 15}] (* _Robert G. Wilson v_, Jan 30 2004 *)
%t A078365 LinearRecurrence[{15,-1},{2,15},20] (* _Harvey P. Dale_, Nov 09 2022 *)
%o A078365 (Sage) [lucas_number2(n,15,1) for n in range(0,20)] # _Zerinvary Lajos_, Jun 26 2008
%Y A078365 a(n)=sqrt(4 + 221*A078364(n-1)^2), n>=1, (Pell equation d=221, +4).
%Y A078365 Cf. A077428, A078355 (Pell +4 equations).
%K A078365 nonn,easy
%O A078365 0,1
%A A078365 _Wolfdieter Lang_, Nov 29 2002