A349072 a(n) = T(n, 3*n), where T(n, x) is the Chebyshev polynomial of the first kind.

1, 3, 71, 2889, 164737, 12082575, 1083358151, 114812765781, 14040770918401, 1946133989077851, 301491888156044999, 51624542295308885793, 9681761035138427706241, 1973656779656041723763559, 434528364117341972641648967, 102755067271708508826774929325

Offset: 0

Author

Vaclav Kotesovec, Nov 07 2021

Keywords

• nonn

Comments

In general, for k>=1, T(n, k*n) ~ 2^(n-1) * k^n * n^n.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..306
• Eric Weisstein's World of Mathematics, Chebyshev Polynomial of the First Kind.
• Wikipedia, Chebyshev polynomials.

Crossrefs

Cf. A053120, A115066, A349070, A349071, A349076.

Programs

Mathematica

Table[ChebyshevT[n, 3*n], {n, 0, 20}]

PARI

a(n) = polchebyshev(n, 1, 3*n); \\ Michel Marcus, Nov 07 2021

Python

from sympy import chebyshevt
def A349072(n): return chebyshevt(n,3*n) # Chai Wah Wu, Nov 08 2023

Formula

a(n) = cosh(n*arccosh(3*n)).

a(n) = ((3*n + sqrt(9*n^2-1))^n + (3*n - sqrt(9*n^2-1))^n)/2.

a(n) ~ 2^(n-1) * 3^n * n^n.