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

1, 33, 1121, 38081, 1293633, 43945441, 1492851361, 50713000833, 1722749176961, 58522759015841, 1988051057361633, 67535213191279681, 2294209197446147521, 77935577499977736033, 2647515425801796877601

Offset

0,2

Comments

(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).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Z. Cerin, G. M. Gianella, On sums of squares of Pell-Lucas Numbers, INTEGERS 6 (2006) #A15

Tanya Khovanova, Recursive Sequences

S. Vidhyalakshmi, V. Krithika, K. Agalya, On The Negative Pell Equation y^2 = 72x^2 - 8, International Journal of Emerging Technologies in Engineering Research (IJETER) Volume 4, Issue 2, February (2016).

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (34,-1).

Formula

a(n) = 34*a(n-1) - a(n-2), a(-1)=1, a(0)=1.

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).

G.f.: (1-x)/(1-34*x+x^2).

a(n) = sqrt(8*A046176(n+1)^2 + 1)/3.

a(n) = (k^n)+(k^(-n))-a(n-1) = A003499(2n)-a(n-1)), where k = (sqrt(2)+1)^4 = 17+12*sqrt(2) and a(0)=1. - Charles L. Hohn, Apr 05 2011

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

Mathematica

LinearRecurrence[{34, -1}, {1, 33}, 20] (* Vincenzo Librandi, Nov 22 2011 *)
a[c_, n_] := Module[{},
   p := Length[ContinuedFraction[ Sqrt[ c]][[2]]];
   d := Denominator[Convergents[Sqrt[c], n p]];
   t := Table[d[[1 + i]], {i, 0, Length[d] - 1, p}];
   Return[t];
] (* Complement of A041027 *)
a[18, 20] (* Gerry Martens, Jun 07 2015 *)

Prog

(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
(PARI) Vec((1-x)/(1-34*x+x^2)+O(x^99)) \\ Charles R Greathouse IV, Nov 22 2011
(Maxima) makelist(expand(((1+sqrt(2))^(4*n+2)+(1-sqrt(2))^(4*n+2))/6), n, 0, 14);  /* _Bruno Berselli, Nov 22 2011 */

Crossrefs

Cf. A056771 (even part).

Row 34 of array A094954.

Row 3 of array A188646.

Cf. similar sequences listed in A238379.

Similar sequences of the type cosh((2*n+1)*arccosh(k))/k are listed in A302329. This is the case k=3.

Sequence in context: A187539 A130835 A262101 * A158688 A353114 A294436

Adjacent sequences: A077417 A077418 A077419 * A077421 A077422 A077423

Keyword

nonn,easy

Author

Wolfdieter Lang, Nov 29 2002

Status

approved