|
%I #27 Jan 15 2022 09:51:12
%S 0,1,1,2,4,9,20,46,106,245,567,1313,3041,7044,16317,37798,87559,
%T 202831,469860,1088436,2521375,5840796,13530276,31343052,72606569,
%U 168194019,389623535,902567761,2090809436,4843386045
%N a(n) = floor( phi*a(n-1) ) + floor( phi*a(n-2) ), where phi is the golden ratio.
%H Harvey P. Dale, <a href="/A005908/b005908.txt">Table of n, a(n) for n = 0..1000</a>
%F Limit_{n -> infinity} a(n)/a(n-1) = phi/C = 2.31651242917313... where phi is the golden ratio and C the positive root of x^4 + x^3 - 2*x^2 - 2*x - 1 (C = 0.6984784404...). - _Benoit Cloitre_, Aug 13 2002
%F Conjectures from _Chai Wah Wu_, Jun 13 2020: (Start)
%F a(n) = 2*a(n-1) + a(n-2) - a(n-4) - a(n-5) for n > 4.
%F G.f.: x*(-x^3 - x^2 - x + 1)/(x^5 + x^4 - x^2 - 2*x + 1). (End)
%t nxt[{a_,b_}]:={b,Floor[GoldenRatio*a]+Floor[GoldenRatio*b]}; Transpose[ NestList[ nxt,{0,1},30]][[1]] (* _Harvey P. Dale_, Apr 24 2016 *)
%Y Cf. A001622 (golden ratio).
%K nonn
%O 0,4
%A _N. J. A. Sloane_
|
