A099156 a(n) = 2^(n-1)*ChebyshevU(n-1, 2).

0, 1, 8, 60, 448, 3344, 24960, 186304, 1390592, 10379520, 77473792, 578272256, 4316282880, 32217174016, 240472260608, 1794909388800, 13397386067968, 99999450988544, 746406063636480, 5571250705137664, 41584381386555392

Offset: 0

Author

Paul Barry, Oct 01 2004

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (8,-4).

Crossrefs

Cf. A090965, A102591.

Programs

Magma

[n le 2 select n-1 else 8*Self(n-1) -4*Self(n-2): n in [1..30]]; // G. C. Greubel, Jul 20 2023

Mathematica

LinearRecurrence[{8,-4}, {0,1}, 40] (* G. C. Greubel, Jul 20 2023 *)

Python

from sympy import chebyshevu
def A099156(n): return chebyshevu(n-1,2)<Chai Wah Wu, Nov 08 2023

SageMath

[lucas_number1(n, 8, 4) for n in range(21)]  # Zerinvary Lajos, Apr 23 2009

SageMath

A099156=BinaryRecurrenceSequence(8,-4,0,1)
[A099156(n) for n in range(41)] # G. C. Greubel, Jul 20 2023

Formula

G.f.: x/(1-8*x+4*x^2).

E.g.f.: exp(4*x) * sinh(2*sqrt(3)*x) / sqrt(3).

a(n) = 8*a(n-1) - 4*a(n-2).

a(n) = sqrt(3) / 12 *( (1+sqrt(3))^(2*n) - (sqrt(3)-1)^(2*n) ).

a(n) = Sum_{k=0..n} binomial(2*n, 2*k+1) * 3^k/2.

a(n) = 2^(n-1)*U(n-1, 4/2) where U is the Chebyshev polynomial of the second kind.

a(n) = 2^(n-1)*A001353(n). - R. J. Mathar, Sep 11 2019

a(n) = 2^(2*n-1)*Sum_{k >= n} binomial(2*k,2*n-1)*(1/3)^(k+1). Cf. A102591. - Peter Bala, Nov 29 2021

a(n+1) = Sum_{i>=0} Sum{j>=0} 2^(2*n-i-j)*binomial(n-i,j)*binomial(n-j,i). - Greg Dresden and Bowen Shi, Aug 28 2023