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A049651 a(n) = (F(3*n+1) - 1)/2, where F=A000045 (the Fibonacci sequence). 7
0, 1, 6, 27, 116, 493, 2090, 8855, 37512, 158905, 673134, 2851443, 12078908, 51167077, 216747218, 918155951, 3889371024, 16475640049, 69791931222, 295643364939, 1252365390980, 5305104928861, 22472785106426 (list; graph; refs; listen; history; text; internal format)
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

Wolfdieter Lang, Notes on certain inhomogeneous three term recurrences. [From Wolfdieter Lang, Oct 18 2010]

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

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified June 24 17:48 EDT 2019. Contains 324326 sequences. (Running on oeis4.)