OFFSET
0,2
COMMENTS
Used for all positive integer solutions of Pell equation x^2 - 53*y^2 = -4. See A097837 with A097838.
a(n-1), with a(-1) := 0, and b(n) := A099368(n) give the proper and improper nonnegative solutions of the Pell equation b(n)^2 - 53*(7*a(n-1))^2 = +4, n >= 0. - Wolfdieter Lang, Jun 27 2013
LINKS
Indranil Ghosh, Table of n, a(n) for n = 0..584
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 linear recurrences with constant coefficients, signature (51,-1).
FORMULA
a(n) = S(n, 51)=U(n, 51/2)= S(2*n+1, sqrt(53))/sqrt(53) 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) = 51*a(n-1) - a(n-2), n >= 1, a(-1)=0, a(0)=1, a(1)=51.
a(n) = (ap^(n+1) - am^(n+1))/(ap-am) with ap := (51+7*sqrt(53))/2 and am := (51-7*sqrt(53))/2 = 1/ap.
G.f.: 1/(1-51*x+x^2).
MATHEMATICA
LinearRecurrence[{51, -1}, {1, 51}, 30] (* G. C. Greubel, Jan 12 2019 *)
PROG
(PARI) my(x='x+O('x^30)); Vec(1/(1-51*x+x^2)) \\ G. C. Greubel, Jan 12 2019
(Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( 1/(1-51*x+x^2) )); // G. C. Greubel, Jan 12 2019
(SageMath) (1/(1-51*x+x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 12 2019
(GAP) a:=[1, 51];; for n in [2..30] do a[n]:=51*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jan 12 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Sep 10 2004
STATUS
approved