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
Zvonko Čerin and Gian Mario Gianella, On sums of squares of Pell-Lucas Numbers, INTEGERS 6 (2006) #A15.
Tanya Khovanova, Recursive Sequences
S. Vidhyalakshmi, V. Krithika, and 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 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(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
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^20)) \\ 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, 17); /* Bruno Berselli, Nov 22 2011 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Nov 29 2002
STATUS
approved