A097838 First differences of Chebyshev polynomials S(n,51) = A097836(n) with Diophantine property.

1, 50, 2549, 129949, 6624850, 337737401, 17217982601, 877779375250, 44749530155149, 2281348258537349, 116304011655249650, 5929223246159194801, 302274081542463685201, 15410048935419488750450, 785610221624851462587749

Offset: 0

Author

Wolfdieter Lang, Sep 10 2004

Keywords

• nonn
• easy

Comments

(7*b(n))^2 - 53*a(n)^2 = -4 with b(n)=A097837(n) give all positive solutions of this Pell equation.

Examples

			All positive solutions of Pell equation x^2 - 53*y^2 = -4 are (7=7*1,1), (364=7*52,50), (18557=7*2651,2549), (946043=7*135149,129949), ...
		

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..584
• 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 (51, -1).
• Index entries for sequences related to Chebyshev polynomials.

Programs

GAP

a:=[1,50];; for n in [3..20] do a[n]:=51*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jan 13 2019

Magma

m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x)/(1-51*x+x^2) )); // G. C. Greubel, Jan 13 2019

Mathematica

LinearRecurrence[{51,-1}, {1,50}, 20] (* G. C. Greubel, Jan 13 2019 *)

PARI

my(x='x+O('x^20)); Vec((1-x)/(1-51*x+x^2)) \\ G. C. Greubel, Jan 13 2019

Sage

((1-x)/(1-51*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Jan 13 2019

Formula

a(n) = ((-1)^n)*S(2*n, 7*i) with the imaginary unit i and the S(n, x) = U(n, x/2) Chebyshev polynomials.

G.f.: (1-x)/(1 - 51*x + x^2).

a(n) = S(n, 51) - S(n-1, 51) = T(2*n+1, sqrt(53)/2)/(sqrt(53)/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) = 51*a(n-1) - a(n-2), a(0)=1, a(1)=50. - Philippe Deléham, Nov 18 2008