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

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

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

Crossrefs

Cf. A033887.

Pairwise sums of A049652.

Sequence in context: A171475 A130019 A196919 * A109114 A080619 A265943

Adjacent sequences: A049648 A049649 A049650 * A049652 A049653 A049654

Keyword

nonn,easy

Author

Clark Kimberling

Status

approved