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

0,3

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).

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

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 *)

Prog

(PARI) a(n)=fibonacci(6*n+3)\32 \\ Charles R Greathouse IV, Oct 07 2016
(Magma) [(Fibonacc9(6*n+3)-2)/32: n in [0..30]]; // G. C. Greubel, Dec 02 2017

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).

Sequence in context: A142549 A049629 A162805 * A163110 A163453 A163968

Adjacent sequences: A049661 A049662 A049663 * A049665 A049666 A049667

Keyword

nonn,easy

Author

Clark Kimberling

Extensions

Chebyshev comments from Wolfdieter Lang, Aug 31 2004

Status

approved