A144139 Chebyshev polynomial of the second kind U(4,n).

1, 5, 209, 1189, 3905, 9701, 20305, 37829, 64769, 104005, 158801, 232805, 330049, 454949, 612305, 807301, 1045505, 1332869, 1675729, 2080805, 2555201, 3106405, 3742289, 4471109, 5301505, 6242501, 7303505, 8494309, 9825089, 11306405

Offset: 0

Author

Vladimir Joseph Stephan Orlovsky, Sep 11 2008

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A057722, A079414, A165900, A193250.

Programs

Magma

[16*n^4-12*n^2+1: n in [0..40]]; // Vincenzo Librandi, May 29 2014

Mathematica

lst={}; Do[AppendTo[lst, ChebyshevU[4, n]], {n, 0, 9^2}]; lst
CoefficientList[Series[(1 + 194 x^2 + 184 x^3 + 5 x^4)/(1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, May 29 2014 *)

Formula

G.f.: (1 + 194*x^2 + 184*x^3 + 5*x^4)/(1 - x)^5. - Vincenzo Librandi, May 29 2014

a(n) = 16*n^4-12*n^2+1 = (4*n^2-2*n-1)*(4*n^2+2*n-1). - Vincenzo Librandi, May 29 2014

From Klaus Purath, Sep 08 2022: (Start)

a(n) = A165900(2*n)*A165900(2*n+1).

a(n) = A057722(2*n).

a(n) = 4*(Sum_{i=1..n} A193250(i)) + 1 = 4*A079414(n) + 1.

(End)

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Wesley Ivan Hurt, Aug 04 2025

Extensions

Changed offset from 1 to 0 by Vincenzo Librandi, May 29 2014