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A056771 a(n) = a(-n) = 34*a(n-1) - a(n-2), and a(0)=1, a(1)=17. 9
1, 17, 577, 19601, 665857, 22619537, 768398401, 26102926097, 886731088897, 30122754096401, 1023286908188737, 34761632124320657, 1180872205318713601, 40114893348711941777, 1362725501650887306817, 46292552162781456490001 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The sequence satisfies the Pell equation a(n)^2 - 18 * A202299(n+1)^2 = 1. - Vincenzo Librandi, Dec 19 2011

Also numbers n such that n - 1 and 2*n + 2 are squares. - Arkadiusz Wesolowski, Mar 15 2015

And they, n - 1 and 2*n + 2, are the squares of A005319 and A003499. - Michel Marcus, Mar 15 2015

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) = (r^n+1/r^n)/2 with r = 17+sqrt(17^2-1).

a(n) = 16*A001110(n)+1 = A001541(2n) = (4*A001109(n))^2+1 = 3*A001109(2n-1)-A001109(2n-2) = A001109(2n)-3*A001109(2n-1).

a(n) = T(n, 17) = T(2*n, 3) with T(n, x) Chebyshev's polynomials of the first kind. See A053120. T(n, 3)= A001541(n).

G.f.: x*(1-17*x)/(1-34*x+x^2).

a(n) = Cosh(2n*ArcSinh(Sqrt(8))). - Herbert Kociemba, Apr 24 2008

a(n) = (a^n + b^n)/2 where a=17+12sqrt(2) and b=17-12sqrt(2). Sqrt(a(n)-1)/4 = A001109(n). - James R. Buddenhagen, Dec 09 2011

a(-n) = a(n). - Michael Somos, May 28 2014

EXAMPLE

G.f. = 1 + 17*x + 577*x^2 + 19601*x^3 + 665857*x^4 + 22619537*x^5 + ...

MATHEMATICA

LinearRecurrence[{34, -1}, {1, 17}, 30] (* Vincenzo Librandi, Dec 18 2011 *)

a[ n_] := ChebyshevT[ 2 n, 3]; (* Michael Somos, May 28 2014 *)

PROG

(Sage) [lucas_number2(n, 34, 1)/2 for n in xrange(0, 15)] # Zerinvary Lajos, Jun 27 2008

(MAGMA) I:=[1, 17]; [n le 2 select I[n] else 34*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Dec 18 2011

(Maxima) makelist(expand(((17+sqrt(288))^n+(17-sqrt(288))^n))/2, n, 0, 15); // Vincenzo Librandi, Dec 18 2011

CROSSREFS

Cf. A001075, A001541, A001091, A001079, A023038, A011943, A001081, A023039, A001085 and note relationship with square triangular number sequences A001110 and A001109.

Row 3 of array A188644

Sequence in context: A012069 A191865 A249862 * A041547 A041544 A202407

Adjacent sequences:  A056768 A056769 A056770 * A056772 A056773 A056774

KEYWORD

nonn,easy

AUTHOR

Henry Bottomley, Aug 16 2000

EXTENSIONS

More terms from James A. Sellers, Sep 07 2000

Chebyshev comments from Wolfdieter Lang, Nov 29 2002

STATUS

approved

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Last modified March 30 14:31 EDT 2017. Contains 284302 sequences.