A322888 Chebyshev T-polynomials T_n(16).

1, 16, 511, 16336, 522241, 16695376, 533729791, 17062657936, 545471324161, 17438019715216, 557471159562751, 17821639086292816, 569734979601807361, 18213697708171542736, 582268591681887560191, 18614381236112230383376, 595077930963909484707841

Offset: 0

Author

Seiichi Manyama, Dec 29 2018

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 0..600
• Wikipedia, Chebyshev polynomials.
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (32, -1).

Crossrefs

Column 16 of A322836.

Programs

GAP

a:=[1,16];; for n in [3..20] do a[n]:=32*a[n-1]-a[n-2]; od; Print(a); # Muniru A Asiru, Dec 31 2018

Magma

I:=[1, 16]; [n le 2 select I[n] else 32*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Jan 02 2019

Maple

seq(coeff(series((1-16*x)/(1-32*x+x^2),x,n+1), x, n), n = 0 .. 20); # Muniru A Asiru, Dec 31 2018

Mathematica

Array[ChebyshevT[#, 16] &, 17, 0] (* or *)
With[{k = 16}, CoefficientList[Series[(1 - k x)/(1 - 2 k x + x^2), {x, 0, 16}], x]] (* Michael De Vlieger, Jan 01 2019 *)

PARI

{a(n) = polchebyshev(n, 1, 16)}

PARI

Vec((1 - 16*x) / (1 - 32*x + x^2) + O(x^20)) \\ Colin Barker, Dec 30 2018

Formula

a(0) = 1, a(1) = 16 and a(n) = 32*a(n-1) - a(n-2) for n > 1.

From Colin Barker, Dec 30 2018: (Start)

G.f.: (1 - 16*x) / (1 - 32*x + x^2).

a(n) = ((16+sqrt(255))^(-n) * (1+(16+sqrt(255))^(2*n))) / 2.

(End)