|
%I #30 Sep 08 2022 08:45:14
%S 1,83,6888,571621,47437655,3936753744,326703123097,27112422463307,
%T 2250004361331384,186723249568041565,15495779709786118511,
%U 1285962992662679794848,106719432611292636853873,8856426943744626179076611,734976716898192680226504840
%N Chebyshev polynomials S(n,83).
%C Used for all positive integer solutions of Pell equation x^2 - 85*y^2 = -4. See A097840 with A097841.
%H Indranil Ghosh, <a href="/A097839/b097839.txt">Table of n, a(n) for n = 0..520</a>
%H Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Nemeth/nemeth7.html">Ellipse Chains and Associated Sequences</a>, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
%H R. Flórez, R. A. Higuita, and A. Mukherjee, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Mukherjee/mukh2.html">Alternating Sums in the Hosoya Polynomial Triangle</a>, Article 14.9.5 Journal of Integer Sequences, Vol. 17 (2014).
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (83,-1).
%F 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).
%F a(n) = 83*a(n-1) - a(n-2), n >= 1, a(-1)=0, a(0)=1, a(1)=83.
%F 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.
%F G.f.: 1/(1-83*x+x^2).
%t CoefficientList[Series[1/(1-83x+x^2),{x,0,20}],x] (* or *) LinearRecurrence[{83,-1},{1,83},20] (* _Harvey P. Dale_, Oct 11 2012 *)
%o (PARI) my(x='x+O('x^20)); Vec(1/(1-83*x+x^2)) \\ _G. C. Greubel_, Jan 13 2019
%o (Magma) m:=20; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( 1/(1-83*x+x^2) )); // _G. C. Greubel_, Jan 13 2019
%o (Sage) (1/(1-83*x+x^2)).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, Jan 13 2019
%o (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
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Sep 10 2004
%E More terms from _Harvey P. Dale_, Oct 11 2012
|
