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A077420 Bisection of Chebyshev sequence T(n,3) (odd part) with Diophantine property. 11
1, 33, 1121, 38081, 1293633, 43945441, 1492851361, 50713000833, 1722749176961, 58522759015841, 1988051057361633, 67535213191279681, 2294209197446147521, 77935577499977736033, 2647515425801796877601 (list; graph; refs; listen; history; text; internal format)
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).

REFERENCES

S Vidhyalakshmi, V Krithika, K Agalya, On The Negative Pell Equation, International Journal of Emerging Technologies in Engineering Research (IJETER), Volume 4, Issue 2, February (2016) www.ijeter.everscience.org,

LINKS

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

Tanya Khovanova, Recursive Sequences

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.

Sequence in context: A187539 A130835 A262101 * A158688 A242492 A065424

Adjacent sequences:  A077417 A077418 A077419 * A077421 A077422 A077423

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Nov 29 2002

STATUS

approved

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Last modified June 24 05:50 EDT 2017. Contains 288697 sequences.