A049651 - OEIS

%I #66 May 25 2024 09:14:47

%S 0,1,6,27,116,493,2090,8855,37512,158905,673134,2851443,12078908,

%T 51167077,216747218,918155951,3889371024,16475640049,69791931222,

%U 295643364939,1252365390980,5305104928861,22472785106426,95196245354567,403257766524696,1708227311453353,7236167012338110

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

%C 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

%C 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

%D A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 24.

%H G. C. Greubel, <a href="/A049651/b049651.txt">Table of n, a(n) for n = 0..1000</a>

%H Hans Koch, <a href="https://arxiv.org/abs/1907.06804">Golden mean renormalization for the almost Mathieu operator and related skew products</a>, arXiv:1907.06804 [math-ph], 2019.

%H Wolfdieter Lang, <a href="/A049651/a049651.pdf"> Notes on certain inhomogeneous three term recurrences.</a>

%H Hermann Stamm-Wilbrandt, <a href="/A049651/a049651_2.svg">6 interlaced bisections</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-3,-1).

%F From _Ralf Stephan_, Jan 23 2003: (Start)

%F a(n) = 4*a(n-1) + a(n-2) + 2, a(0)=0, a(1)=1.

%F G.f.: x*(1+x)/((1-x)*(1-4*x-x^2)).

%F a(n) is asymptotic to -1/2+(sqrt(5)+5)/20*(sqrt(5)+2)^n. (End)

%F a(n+1) = F(2) + F(5) + F(8) + ... + F(3n+2).

%F 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

%F 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

%F 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

%t (Fibonacci[Range[1,5!,3]]-1)/2 (* _Vladimir Joseph Stephan Orlovsky_, May 18 2010 *)

%t LinearRecurrence[{5, -3, -1}, {0, 1, 6}, 50] (* _G. C. Greubel_, Dec 05 2017 *)

%o (PARI) vector(30,n,n--;(fibonacci(3*n+1) -1)/2) \\ _G. C. Greubel_, Dec 05 2017

%o (Magma) [(Fibonacci(3*n+1) - 1)/2: n in [0..30]]; // _G. C. Greubel_, Dec 05 2017

%o (Sage) [(fibonacci(3*n+1)-1)/2 for n in (0..30)] # _G. C. Greubel_, Apr 19 2019

%Y Cf. A000045, A033887, A077259, A214973.

%Y Pairwise sums of A049652.

%K nonn,easy

%O 0,3

%A _Clark Kimberling_