OFFSET
0,2
COMMENTS
(21*b(n))^2 - 445*a(n)^2 = -4 with b(n)=A098255(n) give all positive solutions of this Pell equation.
LINKS
Indranil Ghosh, Table of n, a(n) for n = 0..377
Tanya Khovanova, Recursive Sequences
Giovanni Lucca, Integer Sequences and Circle Chains Inside a Hyperbola, Forum Geometricorum (2019) Vol. 19, 11-16.
Index entries for linear recurrences with constant coefficients, signature (443, -1).
FORMULA
G.f.: (1 - x)/(1 - 443*x + x^2).
a(n) = ((-1)^n)*S(2*n, 21*i) with the imaginary unit i and the S(n, x)=U(n, x/2) Chebyshev polynomials.
a(n) = S(n, 443) - S(n-1, 443) = T(2*n+1, sqrt(445)/2)/(sqrt(445)/2), with S(n, x)=U(n, x/2) Chebyshev's polynomials of the second kind, A049310. S(-1, x)= 0 = U(-1, x) and T(n, x) Chebyshev's polynomials of the first kind, A053120.
a(n) = 443*a(n-1) - a(n-2) for n>1, a(0)=1, a(1)=442. - Philippe Deléham, Nov 18 2008
EXAMPLE
All positive solutions of Pell equation x^2 - 445*y^2 = -4 are (21=21*1,1), (9324=21*444,442), (4130511=21*196691,195805), (1829807049=21*87133669,86741173), ...
MATHEMATICA
LinearRecurrence[{443, -1}, {1, 442}, 20] (* G. C. Greubel, Aug 01 2019 *)
PROG
(PARI) my(x='x+O('x^20)); Vec((1-x)/(1-443*x+x^2)) \\ G. C. Greubel, Aug 01 2019
(Magma) I:=[1, 442]; [n le 2 select I[n] else 443*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Aug 01 2019
(Sage) ((1-x)/(1-443*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019
(GAP) a:=[1, 442];; for n in [3..20] do a[n]:=443*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Sep 10 2004
STATUS
approved