A049664 a(n) = (F(6*n+3) - 2)/32, where F = A000045 (the Fibonacci sequence).

0, 1, 19, 342, 6138, 110143, 1976437, 35465724, 636406596, 11419853005, 204920947495, 3677157201906, 65983908686814, 1184033199160747, 21246613676206633, 381255012972558648, 6841343619829849032, 122762930143964723929, 2202891398971535181691

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• easy

Comments

Partial sums of Chebyshev polynomials S(n,18).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..750
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (19,-19,1).

Crossrefs

Cf. A000045, A001622, A049660, A053606.

Cf. A212336 for more sequences with g.f. of the type 1/(1-k*x+k*x^2-x^3).

Programs

Magma

[(Fibonacc9(6*n+3)-2)/32: n in [0..30]]; // G. C. Greubel, Dec 02 2017

Mathematica

LinearRecurrence[{19, -19, 1}, {0, 1, 19}, 50] (* or *) Table[(Fibonacci[ 6*n +3] - 2)/32, {n,0,30}] (* G. C. Greubel, Dec 02 2017 *)

PARI

a(n)=fibonacci(6*n+3)\32 \\ Charles R Greathouse IV, Oct 07 2016

Formula

G.f.: x/(1-19*x+19*x^2-x^3) = x/((1-x)*(1-18*x+x^2)).

a(n+1) = Sum_{k=0..n} S(k, 18), with n>=0, S(k, 18) = U(k, 9) = A049660(k+1).

a(n) = 19*a(n-1) - 19*a(n-2) + a(n-3), n>=3, a(0)=0, a(1)=1, a(2)=19.

a(n) = 18*a(n-1) - a(n-2) + 1, n>=2, a(0)=0, a(1)=1.

a(n+1) = (S(n+1, 18) - S(n, 18) - 1)/16, n>=0.

a(n) = (1/8)*Sum_{k=0..n} Fibonacci(6*k). - Gary Detlefs, Dec 07 2010

Product_{n>=2} (1 - 1/a(n)) = phi^6/19 = (4*sqrt(5)+9)/19, where phi is the golden ratio (A001622). - Amiram Eldar, Nov 28 2024

Extensions

Chebyshev comments from Wolfdieter Lang, Aug 31 2004