A322889 Chebyshev T-polynomials T_n(18).

1, 18, 647, 23274, 837217, 30116538, 1083358151, 38970776898, 1401864610177, 50428155189474, 1814011722210887, 65253993844402458, 2347329766676277601, 84438617606501591178, 3037442904067381004807, 109263505928819214581874

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 (36,-1).

Crossrefs

Column 18 of A322836.

Programs

GAP

a:=[1,18];; for n in [3..20] do a[n]:=36*a[n-1]-a[n-2]; od; Print(a); # Muniru A Asiru, Dec 31 2018

Magma

I:=[1, 18]; [n le 2 select I[n] else 36*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Jan 02 2019

Maple

seq(coeff(series((1-18*x)/(1-36*x+x^2),x,n+1), x, n), n = 0 .. 20); # Muniru A Asiru, Dec 31 2018

Mathematica

Array[ChebyshevT[#, 18] &, 16, 0] (* or *)
With[{k = 18}, CoefficientList[Series[(1 - k x)/(1 - 2 k x + x^2), {x, 0, 15}], x]] (* Michael De Vlieger, Jan 01 2019 *)

PARI

{a(n) = polchebyshev(n, 1, 18)}

PARI

Vec((1 - 18*x) / (1 - 36*x + x^2) + O(x^20)) \\ Colin Barker, Dec 30 2018

Formula

a(0) = 1, a(1) = 18 and a(n) = 36*a(n-1) - a(n-2) for n > 1.

From Colin Barker, Dec 30 2018: (Start)

G.f.: (1 - 18*x) / (1 - 36*x + x^2).

a(n) = ((18+sqrt(323))^(-n) * (1+(18+sqrt(323))^(2*n))) / 2. (End)

E.g.f.: exp(18*x)*cosh(sqrt(323)*x). - Stefano Spezia, Aug 02 2025