A254745 Chebyshev polynomials of the second kind, U(n,x)^2, evaluated at x = sqrt(3)/2.

1, 3, 4, 3, 1, 0, 1, 3, 4, 3, 1, 0, 1, 3, 4, 3, 1, 0, 1, 3, 4, 3, 1, 0, 1, 3, 4, 3, 1, 0, 1, 3, 4, 3, 1, 0, 1, 3, 4, 3, 1, 0, 1, 3, 4, 3, 1, 0, 1, 3, 4, 3, 1, 0, 1, 3, 4, 3, 1, 0, 1, 3, 4, 3, 1, 0, 1, 3, 4, 3, 1, 0, 1, 3, 4, 3, 1, 0, 1, 3, 4, 3, 1, 0, 1, 3, 4

Offset: 0

Author

Michael Somos, Feb 07 2015

Keywords

• nonn
• easy

Comments

Period 6: repeat [1, 3, 4, 3, 1, 0].

Examples

			G.f. = 1 + 3*x + 4*x^2 + 3*x^3 + x^4 + x^6 + 3*x^7 + 4*x^8 + 3*x^9 + ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,-2,1).

Crossrefs

Cf. A078070, A131027, A254744, A254612, A254707, A254708, A254875.

Programs

Magma

m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x)/((1-x)*(1-x+x^2)))); // G. C. Greubel, Aug 03 2018

Mathematica

a[ n_] := {3, 4, 3, 1, 0, 1}[[Mod[n, 6, 1]]];
a[ n_] := ChebyshevU[ n, Sqrt[3] / 2]^2;
CoefficientList[Series[(1 + x) / ((1 - x) (1 - x + x^2)), {x, 0, 100}], x] (* Vincenzo Librandi, Jul 14 2017 *)

PARI

{a(n) = [1, 3, 4, 3, 1, 0][n%6 + 1]};

PARI

{a(n) = simplify( polchebyshev( n, 2, quadgen(12) / 2)^2)};

Formula

Euler transform of length 6 sequence [3, -2, -1, 0, 0, 1].

G.f.: (1 + x) / ((1 - x) * (1 - x + x^2)) = (1 - x^2)^2 * (1 - x^3) / ((1 - x)^3 * (1 - x^6)).

a(n) = a(-2-n) = a(n+6) for all n in Z.

a(n) = (-1)^n*A078070(n) = A131027(n-1) for all n in Z.

a(n) = (n+1)*(Sum_{k=0..n} (-1)^k/(k+1)*binomial(n+k+1,2*k+1)) for n >= 0. - Werner Schulte, Jul 10 2017

Sum_{n>=0} a(n)/(n+1)*x^(n+1) = log(1-x+x^2)-2*log(1-x) for -1 < x < 1. - Werner Schulte, Jul 10 2017

a(n) = sqrt(3)*sin(Pi*n/3) - cos(Pi*n/3) + 2. - Peter Luschny, Jul 16 2017

a(n) = 2 + 2*cos(Pi/3*(n+4)) for n >= 0. - Werner Schulte, Jul 18 2017