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%I #40 Sep 08 2022 08:45:49
%S 1,2,3,5,7,12,18,30,47,77,123,200,322,522,843,1365,2207,3572,5778,
%T 9350,15127,24477,39603,64080,103682,167762,271443,439205,710647,
%U 1149852,1860498,3010350,4870847,7881197,12752043,20633240,33385282
%N Ceiling(phi^n) where phi = (1+sqrt(5))/2.
%H Danny Rorabaugh, <a href="/A169986/b169986.txt">Table of n, a(n) for n = 0..4000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-1,-1).
%F For n >= 5, a(n) = a(n-1) + 2a(n-2) - a(n-3) - a(n-4). - _Charles R Greathouse IV_, Oct 14 2010
%F a(n) = 3*Fibonacci(n-1) + Fibonacci(n-2) + (n mod 2), n>0. - _Gary Detlefs_, Dec 29 2010
%F G.f.: (-x+x^2+x^3+x^4-1) / ((1-x)*(1+x)*(x^2+x-1)). - _R. J. Mathar_, Jan 06 2011
%F a(2k) = A000032(2k) = A169985(2k) and a(2k+1) = A000032(2k+1)+1 = A169985(2k+1)+1, for k>0. - _Danny Rorabaugh_, Apr 15 2015
%t Ceiling[GoldenRatio^Range[0,40]] (* or *) Join[{1},LinearRecurrence[{1,2,-1,-1},{2,3,5,7},40]] (* _Harvey P. Dale_, Nov 12 2014 *)
%o (Magma) [1] cat [3*Fibonacci(n-1) + Fibonacci(n-2)+ n mod 2: n in [1..40]]; // _Vincenzo Librandi_, Apr 16 2015
%o (Sage) [ceil(golden_ratio^n) for n in range(37)] # _Danny Rorabaugh_, Apr 16 2015
%o (PARI) a(n)=if(n, 3*fibonacci(n-1) + fibonacci(n-2) + n%2, 1) \\ _Charles R Greathouse IV_, Apr 16 2015
%Y Cf. A001622, A014217, A062724, A169985.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Sep 26 2010