OFFSET
0,3
COMMENTS
This is the sequence A(0,1;4,1;2) of the family of sequences [a,b:c,d:k] considered by G. Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. - Wolfdieter Lang, Oct 18 2010
For n>0, a(n) is the least number whose greedy Fibonacci-union-Lucas representation (as at A214973), has n terms. - Clark Kimberling, Oct 23 2012
REFERENCES
A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 24.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Hans Koch, Golden mean renormalization for the almost Mathieu operator and related skew products, arXiv:1907.06804 [math-ph], 2019.
Wolfdieter Lang, Notes on certain inhomogeneous three term recurrences.
Hermann Stamm-Wilbrandt, 6 interlaced bisections
Index entries for linear recurrences with constant coefficients, signature (5,-3,-1).
FORMULA
From Ralf Stephan, Jan 23 2003: (Start)
a(n) = 4*a(n-1) + a(n-2) + 2, a(0)=0, a(1)=1.
G.f.: x*(1+x)/((1-x)*(1-4*x-x^2)).
a(n) is asymptotic to -1/2+(sqrt(5)+5)/20*(sqrt(5)+2)^n. (End)
a(n+1) = F(2) + F(5) + F(8) + ... + F(3n+2).
a(n) = 5*a(n-1) - 3*a(n-2) - a(n-3), a(0)=0, a(1)=1, a(2)= 6. Observation by G. Detlefs. See the W. Lang link. - Wolfdieter Lang, Oct 18 2010
a(2*n) = A077259(2*n); a(2*n+1) = A294262(2*n+1). See "6 interlaced bisections" link. - Hermann Stamm-Wilbrandt, Apr 18 2019
E.g.f.: exp(x)*(exp(x)*(5*cosh(sqrt(5)*x) + sqrt(5)*sinh(sqrt(5)*x)) - 5)/10. - Stefano Spezia, May 24 2024
MATHEMATICA
(Fibonacci[Range[1, 5!, 3]]-1)/2 (* Vladimir Joseph Stephan Orlovsky, May 18 2010 *)
LinearRecurrence[{5, -3, -1}, {0, 1, 6}, 50] (* G. C. Greubel, Dec 05 2017 *)
PROG
(PARI) vector(30, n, n--; (fibonacci(3*n+1) -1)/2) \\ G. C. Greubel, Dec 05 2017
(Magma) [(Fibonacci(3*n+1) - 1)/2: n in [0..30]]; // G. C. Greubel, Dec 05 2017
(Sage) [(fibonacci(3*n+1)-1)/2 for n in (0..30)] # G. C. Greubel, Apr 19 2019
KEYWORD
nonn,easy
AUTHOR
STATUS
approved
