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A097839
Chebyshev polynomials S(n,83).
5
1, 83, 6888, 571621, 47437655, 3936753744, 326703123097, 27112422463307, 2250004361331384, 186723249568041565, 15495779709786118511, 1285962992662679794848, 106719432611292636853873, 8856426943744626179076611, 734976716898192680226504840
OFFSET
0,2
COMMENTS
Used for all positive integer solutions of Pell equation x^2 - 85*y^2 = -4. See A097840 with A097841.
LINKS
Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
R. Flórez, R. A. Higuita, and A. Mukherjee, Alternating Sums in the Hosoya Polynomial Triangle, Article 14.9.5 Journal of Integer Sequences, Vol. 17 (2014).
Tanya Khovanova, Recursive Sequences
FORMULA
a(n) = S(n, 83) = U(n, 83/2) = S(2*n+1, sqrt(85))/sqrt(85) 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) = 83*a(n-1) - a(n-2), n >= 1, a(-1)=0, a(0)=1, a(1)=83.
a(n) = (ap^(n+1) - am^(n+1))/(ap - am) with ap = (83+9*sqrt(85))/2 and am = (83-9*sqrt(85))/2 = 1/ap.
G.f.: 1/(1-83*x+x^2).
MATHEMATICA
CoefficientList[Series[1/(1-83x+x^2), {x, 0, 20}], x] (* or *) LinearRecurrence[{83, -1}, {1, 83}, 20] (* Harvey P. Dale, Oct 11 2012 *)
PROG
(PARI) my(x='x+O('x^20)); Vec(1/(1-83*x+x^2)) \\ G. C. Greubel, Jan 13 2019
(Magma) m:=20; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( 1/(1-83*x+x^2) )); // G. C. Greubel, Jan 13 2019
(Sage) (1/(1-83*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Jan 13 2019
(GAP) a:=[1, 83];; for n in [3..20] do a[n]:=83*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jan 13 2019
CROSSREFS
Sequence in context: A252812 A202657 A180846 * A268987 A087189 A201727
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Sep 10 2004
EXTENSIONS
More terms from Harvey P. Dale, Oct 11 2012
STATUS
approved