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%I #43 Sep 08 2022 08:45:50
%S 0,1,2,3,5,9,14,23,38,61,99,161,260,421,682,1103,1785,2889,4674,7563,
%T 12238,19801,32039,51841,83880,135721,219602,355323,574925,930249,
%U 1505174
%N Floor(Lucas(n+1)/2), Lucas(n) = A000032(n).
%C Sequences of the form a(0)=1, a(1)=b,
%C a(n) = a(n-1) + a(n-2) + 1 if n mod 3 =2, else
%C a(n) = a(n-1) + a(n-2) have a closed form of
%C a(n) = F(n-1)*a + F(n)*b + floor(F(n+1)/2),
%C where F(n)= Fibonacci(n) = A000045(n), floor(F(n+1)/2) = A004695(n+1).
%C We can generalize the definition of this sequence by changing the added 1 to any value of k and changing the last term of the formula to floor(F(n+1)/2)*k.
%C Two variants: if we add the constant at n mod 3 = 0, then a(n)=F(n-1)*a + F(n)*b + floor(F(n)/2), and if for n mod 3 =1, then a(n)=F(n-1)*a + F(n)*b + floor(F(n-1)/2).
%H Vincenzo Librandi, <a href="/A173714/b173714.txt">Table of n, a(n) for n = 0..280</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1,-1,-1).
%F a(0)= 0, a(1)=1, a(n)=a(n-1)+a(n-2)+1 if n mod 3 =2, else a(n)=a(n-1)+a(n-2).
%F G.f.: x*(1+x-x^3)/[(1-x-x^2)*(1-x^3)].
%F a(n) = a(n-1) +a(n-2) +(1+(-1)^Fib(n+1))/2.
%F a(n) = A000204(n+1)/2 + A099837(n+1)/6 - 1/3. - _R. J. Mathar_, Nov 26 2010
%F a(n) = Fibonacci(n) + floor(Fibonacci(n+1)/2). - _Gary Detlefs_, Dec 10 2010
%e a(5) = a(4) + a(3) + 1 = 5 +3 +1 =9 because 5 mod 3 = 2.
%e a(6) = a(5) + a(4) = 9 +5 =14 because 6 mod 3 <>2.
%p with(combinat):
%p g:=(a,b,n)->fibonacci(n-1)*a+fibonacci(n)*b + floor(fibonacci(n+1)/2):
%p seq(g(0,1,n),n=0..30)
%t Table[Floor[LucasL[n + 1]/2], {n,0,50}] (* _G. C. Greubel_, Nov 24 2016 *)
%o (Magma) [Floor(Lucas(n+1)/2): n in [0..50]]; // _Vincenzo Librandi_, Apr 24 2011
%K nonn,easy
%O 0,3
%A _Gary Detlefs_, Nov 25 2010
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