A097841 First differences of Chebyshev polynomials S(n,83) = A097839(n) with Diophantine property.

1, 82, 6805, 564733, 46866034, 3889316089, 322766369353, 26785719340210, 2222891938868077, 184473245206710181, 15309056460218076946, 1270467212952893676337, 105433469618629957059025

Offset: 0

Author

Wolfdieter Lang, Sep 10 2004

Keywords

• nonn
• easy

Comments

(9*b(n))^2 - 85*a(n)^2 = -4 with b(n)=A097840(n) give all positive solutions of this Pell equation.

For n > 0, a(n) is the hypotenuse of the Pythagorean triple (x(n), y(n), a(n)) that is primitive for n == 0, 2 (mod 3) where (x(n)) and (y(n)) are recurrences of the form (82,82,-1) with the initial values x(0) = 1, x(1) = 80, x(2) = 6643; y(0) = 0, y(1) = 18, y(2) = 1476. - Klaus Purath, Jul 19 2025

Examples

			All positive solutions of Pell equation x^2 - 85*y^2 = -4 are (9=9*1,1), (756=9*84,82), (62739=9*6971,6805), (5206581=9*578509,564733), ...
		

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..520
• 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 (83, -1).
• Index entries for sequences related to Chebyshev polynomials.

Crossrefs

Cf. A097839, A097840.

Cf. A049310, A053120.

Cf. A099371.

Programs

GAP

a:=[1,82];; for n in [3..20] do a[n]:=83*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)/(1-83*x+x^2) )); // G. C. Greubel, Jan 13 2019

Mathematica

CoefficientList[Series[(1-x)/(1-83x+x^2), {x, 0, 20}], x] (* Michael De Vlieger, Feb 08 2017 *)
LinearRecurrence[{83,-1}, {1,82}, 20] (* G. C. Greubel, Jan 13 2019 *)

PARI

my(x='x+O('x^20)); Vec((1-x)/(1-83*x+x^2)) \\ G. C. Greubel, Jan 13 2019

Sage

((1-x)/(1-83*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Jan 13 2019

Formula

a(n) = ((-1)^n)*S(2*n, 9*i) with the imaginary unit i and the S(n, x) = U(n, x/2) Chebyshev polynomials.

G.f.: (1-x)/(1 - 83*x + x^2).

a(n) = S(n, 83) - S(n-1, 83) = T(2*n+1, sqrt(85)/2)/(sqrt(85)/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) = 83*a(n-1) - a(n-2) for n > 1, a(0)=1, a(1)=82. - Philippe Deléham, Nov 18 2008

From Klaus Purath, Jul 19 2025: (Start)

a(n) = A099371(2n+1) = A099371(n)^2 + A099371(n+1)^2.

a(n) = (t(i+2*n+1) + t(i))/(t(i+n+1) + t(i+n)) as long as t(i+n+1) + t(i+n) != 0 for any integer i and n >= 1 where (t) is a sequence satisfying t(i+3) = 82*t(i+2) + 82*t(i+1) - t(i) or t(i+2) = 83*t(i+1) - t(i) regardless of initial values and including this sequence itself. (End)