|
%I #29 Feb 14 2023 18:13:05
%S 0,0,1,2,3,4,6,8,11,14,19,24,32,40,53,66,87,108,142,176,231,286,375,
%T 464,608,752,985,1218,1595,1972,2582,3192,4179,5166,6763,8360,10944,
%U 13528,17709,21890,28655,35420,46366,57312,75023,92734,121391,150048,196416
%N Partial sums of repeated Fibonacci sequence.
%C Equals row sums of triangle A139147 starting with "1". - _Gary W. Adamson_, Apr 11 2008
%H G. C. Greubel, <a href="/A094707/b094707.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1,1,-1).
%F G.f. : x^2*(1+x)/((1-x)*(1-x^2-x^4)).
%F a(n) = a(n-1) + a(n+2) - a(n-3) + a(n-4) - a(n-5).
%F a(n) = Sum_{k=0..n} Fibonacci(floor(k/2)).
%F a(n) = -2 - (sqrt(5)/2 - 1/2)^(n/2)*((2*sqrt(5)/5 - 1)*cos(Pi*n/2) + sqrt(4*sqrt(5)/5 - 8/5)*sin(Pi*n/2)) - (sqrt(5)/2 + 1/2)^(n/2)*((sqrt(sqrt(5)/5 + 2/5) - sqrt(5)/5 - 1/2)*(-1)^n - sqrt(sqrt(5)/5 + 2/5) - sqrt(5)/5-1/2).
%F a(n) = A131524(n) + A131524(n+1). - _R. J. Mathar_, Jul 07 2011
%F a(n) = Fibonacci(n/2 +3) - 2 if n even, otherwise a(n) = 2*Fibonacci((n-1)/2 + 2) - 2. - _G. C. Greubel_, Feb 12 2023
%t LinearRecurrence[{1,1,-1,1,-1}, {0,0,1,2,3}, 50] (* _Jean-François Alcover_, Nov 18 2017 *)
%o (Magma) [Fibonacci(Floor((n+6)/2))*((n+1) mod 2) + 2*Fibonacci(Floor((n+3)/2))*(n mod 2) - 2: n in [0..60]]; // _G. C. Greubel_, Feb 12 2023
%o (SageMath)
%o def A094707(n): return fibonacci((n+6)//2) - 2 if (n%2==0) else 2*fibonacci((n+3)//2) - 2
%o [A094707(n) for n in range(61)] # _G. C. Greubel_, Feb 12 2023
%Y Cf. A000045, A103609, A131524, A139147.
%K easy,nonn
%O 0,4
%A _Paul Barry_, May 21 2004
|
