A049651 a(n) = (F(3*n+1) - 1)/2, where F=A000045 (the Fibonacci sequence).

0, 1, 6, 27, 116, 493, 2090, 8855, 37512, 158905, 673134, 2851443, 12078908, 51167077, 216747218, 918155951, 3889371024, 16475640049, 69791931222, 295643364939, 1252365390980, 5305104928861, 22472785106426, 95196245354567, 403257766524696, 1708227311453353, 7236167012338110

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A000045, A033887, A077259, A214973.

Pairwise sums of A049652.

Programs

Magma

[(Fibonacci(3*n+1) - 1)/2: n in [0..30]]; // G. C. Greubel, Dec 05 2017

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 *)

PARI

vector(30,n,n--;(fibonacci(3*n+1) -1)/2) \\ G. C. Greubel, Dec 05 2017

Sage

[(fibonacci(3*n+1)-1)/2 for n in (0..30)] # G. C. Greubel, Apr 19 2019

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