cp's OEIS Frontend

A099571 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+3, k).

%I A099571 #12 Jul 25 2022 18:59:54
%S A099571 1,1,5,6,17,23,50,73,138,211,370,581,979,1560,2575,4135,6755,10890,
%T A099571 17700,28590,46356,74946,121380,196326,317797,514123,832025,1346148,
%U A099571 2178293,3524441,5702870,9227311,14930334,24157645,39088150,63245795
%N A099571 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+3, k).
%C A099571 Fourth column of triangle A054450.
%H A099571 G. C. Greubel, <a href="/A099571/b099571.txt">Table of n, a(n) for n = 0..1000</a>
%H A099571 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,-3,-6,3,4,-1,-1).
%F A099571 G.f.: 1/((1-x^2)^3*(1-x-x^2)).
%F A099571 a(n) = a(n-1) + 4*a(n-2) - 3*a(n-3) - 6*a(n-4) + 3*a(n-5) + 4*a(n-6) - a(n-7) - a(n-8);
%F A099571 a(n) = Sum_{k=0..n} Fibonacci(n-k+1)*binomial(k/2 +2, 2)*(1+(-1)^k)/2.
%F A099571 a(n) = Fibonacci(n+4) + (-1)^n*(n^2 + 4*n + 7)/16 - (n^2 + 12*n + 39)/16. - _G. C. Greubel_, Jul 25 2022
%t A099571 Table[Sum[Binomial[n-k+3,k],{k,0,Floor[n/2]}],{n,0,40}] (* or *) LinearRecurrence[{1,4,-3,-6,3,4,-1,-1},{1,1,5,6,17,23,50,73},40] (* _Harvey P. Dale_, Jun 04 2021 *)
%o A099571 (Magma) [Fibonacci(n+4) +(-1)^n*(n^2+4*n+7)/16 -(n^2+12*n+39)/16: n in [0..40]]; // _G. C. Greubel_, Jul 25 2022
%o A099571 (SageMath) [fibonacci(n+4) +(-1)^n*(n^2+4*n+7)/16 -(n^2+12*n+39)/16 for n in (0..40)] # _G. C. Greubel_, Jul 25 2022
%Y A099571 Cf. A000045, A052952, A054450, A054451, A099572.
%K A099571 easy,nonn
%O A099571 0,3
%A A099571 _Paul Barry_, Oct 23 2004