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%I #34 Aug 23 2023 07:19:21
%S 2,17,287,4862,82367,1395377,23639042,400468337,6784322687,
%T 114933017342,1947076972127,32985375508817,558804306677762,
%U 9466687838013137,160374888939545567,2716906424134261502,46027034321342899967,779742677038695037937,13209598475336472744962
%N A Chebyshev T-sequence with Diophantine property.
%C a(n) gives the general (positive integer) solution of the Pell equation a^2 - 285*b^2 =+4 with companion sequence b(n)=A078366(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 Stefano Spezia, <a href="/A078367/b078367.txt">Table of n, a(n) for n = 0..800</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 (17,-1).
%F a(n) = 17*a(n-1)-a(n-2), n >= 1; a(-1)=17, a(0)=2.
%F a(n) = sqrt(4 + 285*A078366(n-1)^2), n>=1, (Pell equation d=285, +4).
%F a(n) = S(n, 17) - S(n-2, 17) = 2*T(n, 17/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 17)=A078366(n). U-, resp. T-, are Chebyshev's polynomials of the second, resp. first, case. See A049310 and A053120.
%F G.f.: (2-17*x)/(1-17*x+x^2).
%F a(n) = ap^n + am^n, with ap := (17+sqrt(285))/2 and am := (17-sqrt(285))/2.
%F E.g.f.: 2*exp(17*x/2)*cosh(sqrt(285)*x/2). - _Stefano Spezia_, Aug 19 2023
%t a[0] = 2; a[1] = 17; a[n_] := 17a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 15}] (* _Robert G. Wilson v_, Jan 30 2004 *)
%o (PARI) a(n)=if(n<0,0,subst(2*poltchebi(n),x,17/2))
%o (Sage) [lucas_number2(n,17,1) for n in range(0,20)] # _Zerinvary Lajos_, Jun 26 2008
%Y Cf. A077428, A078355 (Pell +4 equations).
%Y Cf. A049310, A053120, A078366.
%K nonn,easy
%O 0,1
%A _Wolfdieter Lang_, Nov 29 2002
%E More terms from _Stefano Spezia_, Aug 19 2023
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