login
A097844
Chebyshev polynomials S(n,171).
4
1, 171, 29240, 4999869, 854948359, 146191169520, 24997835039561, 4274483600595411, 730911697866775720, 124981625851618052709, 21371127108928820237519, 3654337754000976642563040, 624870384807058077058042321, 106849181464252930200282673851
OFFSET
0,2
COMMENTS
Used for all positive integer solutions of Pell equation x^2 - 173*y^2 = -4. See A097845 with A098244.
LINKS
Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
Tanya Khovanova, Recursive Sequences
FORMULA
a(n) = S(n, 171) = U(n, 171/2) = S(2*n+1, sqrt(173))/sqrt(173) with S(n, x) = U(n, x/2) Chebyshev's polynomials of the second kind, A049310. S(-1, x) = 0 = U(-1, x).
a(n) = 171*a(n-1) - a(n-2), n >= 1, a(-1)=0, a(0)=1, a(1)=171.
a(n) = (ap^(n+1) - am^(n+1))/(ap-am) with ap = (171+13*sqrt(173))/2 and am = (171-13*sqrt(173))/2 = 1/ap.
G.f.: 1/(1-171*x+x^2).
MATHEMATICA
CoefficientList[Series[1/(1-171x+x^2), {x, 0, 30}], x] (* or *) LinearRecurrence[{171, -1}, {1, 171}, 30] (* Harvey P. Dale, Mar 21 2013 *)
PROG
(PARI) my(x='x+O('x^30)); Vec(1/(1-171*x+x^2)) \\ G. C. Greubel, Jan 14 2019
(Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( 1/(1-171*x+x^2) )); // G. C. Greubel, Jan 14 2019
(Sage) (1/(1-171*x+x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 14 2019
(GAP) a:=[1, 171];; for n in [3..30] do a[n]:=171*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jan 14 2019
CROSSREFS
Sequence in context: A046166 A262113 A145625 * A076573 A015356 A259158
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Sep 10 2004
STATUS
approved