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A077420 Bisection of Chebyshev sequence T(n,3) (odd part) with Diophantine property.

%I A077420 #54 Aug 22 2024 09:17:45
%S A077420 1,33,1121,38081,1293633,43945441,1492851361,50713000833,
%T A077420 1722749176961,58522759015841,1988051057361633,67535213191279681,
%U A077420 2294209197446147521,77935577499977736033,2647515425801796877601
%N A077420 Bisection of Chebyshev sequence T(n,3) (odd part) with Diophantine property.
%C A077420 (3*a(n))^2 - 2*(2*b(n))^2 = 1 with companion sequence b(n)= A046176(n+1), n>=0 (special solutions of Pell equation).
%H A077420 Vincenzo Librandi, <a href="/A077420/b077420.txt">Table of n, a(n) for n = 0..200</a>
%H A077420 Z. Cerin and G. M. Gianella, <a href="https://eudml.org/doc/126317">On sums of squares of Pell-Lucas Numbers</a>, INTEGERS 6 (2006) #A15
%H A077420 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A077420 S. Vidhyalakshmi, V. Krithika, and K. Agalya, <a href="http://www.ijeter.everscience.org/Manuscripts/Volume-4/Issue-2/Vol-4-issue-2-M-04.pdf">On The Negative Pell Equation y^2 = 72x^2 - 8</a>, International Journal of Emerging Technologies in Engineering Research (IJETER) Volume 4, Issue 2, February (2016).
%H A077420 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A077420 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (34,-1).
%F A077420 a(n) = 34*a(n-1) - a(n-2), a(-1)=1, a(0)=1.
%F A077420 a(n) = T(2*n+1, 3)/3 = S(n, 34) - S(n-1, 34), with S(n, x) := U(n, x/2), resp. T(n, x), Chebyshev's polynomials of the second, resp. first, kind. See A049310 and A053120. S(-1, x)=0, S(n, 34)= A029547(n), T(n, 3)=A001541(n).
%F A077420 G.f.: (1-x)/(1-34*x+x^2).
%F A077420 a(n) = sqrt(8*A046176(n+1)^2 + 1)/3.
%F A077420 a(n) = (k^n)+(k^(-n))-a(n-1) = A003499(2*n)-a(n-1), where k = (sqrt(2)+1)^4 = 17+12*sqrt(2) and a(0)=1. - _Charles L. Hohn_, Apr 05 2011
%F A077420 a(n) = a(-n-1) = A029547(n)-A029547(n-1) = ((1+sqrt(2))^(4n+2)+(1-sqrt(2))^(4n+2))/6. - _Bruno Berselli_, Nov 22 2011
%t A077420 LinearRecurrence[{34,-1},{1,33},20] (* _Vincenzo Librandi_, Nov 22 2011 *)
%t A077420 a[c_, n_] := Module[{},
%t A077420    p := Length[ContinuedFraction[ Sqrt[ c]][[2]]];
%t A077420    d := Denominator[Convergents[Sqrt[c], n p]];
%t A077420    t := Table[d[[1 + i]], {i, 0, Length[d] - 1, p}];
%t A077420    Return[t];
%t A077420 ] (* Complement of A041027 *)
%t A077420 a[18, 20] (* _Gerry Martens_, Jun 07 2015 *)
%o A077420 (Magma) I:=[1,33]; [n le 2 select I[n] else 34*Self(n-1)-Self(n-2): n in [1..20]]; // _Vincenzo Librandi_, Nov 22 2011
%o A077420 (PARI) Vec((1-x)/(1-34*x+x^2)+O(x^99)) \\ _Charles R Greathouse IV_, Nov 22 2011
%o A077420 (Maxima) makelist(expand(((1+sqrt(2))^(4*n+2)+(1-sqrt(2))^(4*n+2))/6),n,0,14);  /* _Bruno Berselli, Nov 22 2011 */
%Y A077420 Cf. A056771 (even part).
%Y A077420 Row 34 of array A094954.
%Y A077420 Row 3 of array A188646.
%Y A077420 Cf. similar sequences listed in A238379.
%Y A077420 Similar sequences of the type cosh((2*n+1)*arccosh(k))/k are listed in A302329. This is the case k=3.
%K A077420 nonn,easy
%O A077420 0,2
%A A077420 _Wolfdieter Lang_, Nov 29 2002