A097839 Chebyshev polynomials S(n,83).
1, 83, 6888, 571621, 47437655, 3936753744, 326703123097, 27112422463307, 2250004361331384, 186723249568041565, 15495779709786118511, 1285962992662679794848, 106719432611292636853873, 8856426943744626179076611, 734976716898192680226504840
Offset: 0
Links
- Indranil Ghosh, Table of n, a(n) for n = 0..520
- 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
- Index entries for sequences related to Chebyshev polynomials.
- Index entries for linear recurrences with constant coefficients, signature (83,-1).
Programs
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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
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Magma
m:=20; R
:=PowerSeriesRing(Integers(), m); Coefficients(R!( 1/(1-83*x+x^2) )); // G. C. Greubel, Jan 13 2019 -
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 *) -
PARI
my(x='x+O('x^20)); Vec(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
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).
Extensions
More terms from Harvey P. Dale, Oct 11 2012
Comments