A049663 a(n) = (F(6*n+5) - 1)/4, where F = A000045 (the Fibonacci sequence).

1, 22, 399, 7164, 128557, 2306866, 41395035, 742803768, 13329072793, 239180506510, 4291920044391, 77015380292532, 1381984925221189, 24798713273688874, 444994854001178547, 7985108658747524976, 143286961003454271025, 2571180189403429353478

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 0..700
• Index entries for linear recurrences with constant coefficients, signature (19,-19,1).

Crossrefs

Cf. A000045, A001622, A049664.

Programs

Magma

[(Fibonacci(6*n+5) - 1)/4: n in [0..30]]; // G. C. Greubel, Dec 02 2017

Mathematica

(Fibonacci[6*Range[0,20]+5]-1)/4 (* or *) LinearRecurrence[{19,-19,1},{1,22,399},20] (* Harvey P. Dale, Sep 22 2016 *)

PARI

Vec((1+3*x)/((1-x)*(1-18*x+x^2)) + O(x^25)) \\ Colin Barker, Mar 04 2016

PARI

for(n=0,30, print1((fibonacci(6*n+5) - 1)/4, ", ")) \\ G. C. Greubel, Dec 02 2017

Formula

From R. J. Mathar, Oct 26 2015: (Start)

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

a(n) = A049664(n+1) + 3*A049664(n). (End)

From Colin Barker, Mar 04 2016: (Start)

a(n) = (-1/4+1/40*(9+4*sqrt(5))^(-n)*(25-11*sqrt(5)+(9+4*sqrt(5))^(2*n)*(25+11*sqrt(5)))).

a(n) = 19*a(n-1) - 19*a(n-2) + a(n-3) for n>2. (End)

Product_{n>=1} (1 - 1/a(n)) = 4*phi^2/11 = 2*(sqrt(5)+3)/11, where phi is the golden ratio (A001622). - Amiram Eldar, Nov 28 2024