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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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Used for all positive integer solutions of Pell equation x^2 - 85*y^2 = -4. See A097840 with A097841. LINKS Indranil Ghosh, Table of n, a(n) for n = 0..520 R. Flórez, R. A. Higuita, 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 linear recurrences with constant coefficients, signature (83, -1). 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:=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 Adjacent sequences:  A097836 A097837 A097838 * A097840 A097841 A097842 KEYWORD nonn,easy AUTHOR Wolfdieter Lang, Sep 10 2004 EXTENSIONS More terms from Harvey P. Dale, Oct 11 2012 STATUS approved

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Last modified October 19 12:09 EDT 2019. Contains 328218 sequences. (Running on oeis4.)