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A078363 A Chebyshev T-sequence with Diophantine property.

%I A078363 #31 Aug 11 2024 23:25:04
%S A078363 2,13,167,2158,27887,360373,4656962,60180133,777684767,10049721838,
%T A078363 129868699127,1678243366813,21687295069442,280256592535933,
%U A078363 3621648407897687,46801172710133998,604793596823844287,7815515585999841733,100996909021174098242
%N A078363 A Chebyshev T-sequence with Diophantine property.
%C A078363 a(n) gives the general (positive integer) solution of the Pell equation a^2 - 165*b^2 = +4 with companion sequence b(n)=A078362(n-1), n>=1.
%C A078363 Except for the first term, positive values of x (or y) satisfying x^2 - 13xy + y^2 + 165 = 0. - _Colin Barker_, Feb 26 2014
%D A078363 O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).
%H A078363 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A078363 <a href="/index/Rea#recur1">Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)</a>
%H A078363 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A078363 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (13,-1).
%F A078363 a(n) = 13*a(n-1)-a(n-2), n >= 1; a(-1)=13, a(0)=2.
%F A078363 a(n) = S(n, 13) - S(n-2, 13) = 2*T(n, 13/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 13)=A078362(n). U-, resp. T-, are Chebyshev's polynomials of the second, resp. first, case. See A049310 and A053120.
%F A078363 G.f.: (2-13*x)/(1-13*x+x^2).
%F A078363 a(n) = ap^n + am^n, with ap := (13+sqrt(165))/2 and am := (13-sqrt(165))/2.
%F A078363 a(n) = sqrt(4 + 165*A078362(n-1)^2), n>=1, (Pell equation d=165, +4).
%F A078363 E.g.f.: 2*exp(13*x/2)*cosh(sqrt(165)*x/2). - _Stefano Spezia_, Sep 24 2022
%t A078363 a[0] = 2; a[1] = 13; a[n_] := 13a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 16}] (* _Robert G. Wilson v_, Jan 30 2004 *)
%t A078363 LinearRecurrence[{13,-1},{2,13},20] (* _Harvey P. Dale_, Oct 28 2016 *)
%o A078363 (PARI) a(n)=if(n<0,0,2*subst(poltchebi(n),x,13/2))
%o A078363 (PARI) a(n)=if(n<0,0,polsym(1-13*x+x^2,n)[n+1])
%o A078363 (PARI) Vec((2-13*x)/(1-13*x+x^2) + O(x^100)) \\ _Colin Barker_, Feb 26 2014
%o A078363 (Sage) [lucas_number2(n,13,1) for n in range(0,20)] # _Zerinvary Lajos_, Jun 25 2008
%Y A078363 Cf. A078362.
%Y A078363 Cf. A077428, A078355 (Pell +4 equations).
%Y A078363 Cf. A049310, A053120.
%K A078363 nonn,easy
%O A078363 0,1
%A A078363 _Wolfdieter Lang_, Nov 29 2002
%E A078363 More terms from _Colin Barker_, Feb 26 2014