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%I #11 Dec 15 2023 12:56:21
%S 0,0,1,5,14,32,67,133,256,484,905,1681,3110,5740,10579,19481,35856,
%T 65976,121377,223277,410702,755432,1389491,2555709,4700720,8646012,
%U 15902537,29249369,53798022,98950036,181997539,334745713
%N a(n) = a(n-1)+a(n-2)+a(n-3)+4*n-8 with a(0)=0, a(1)=0 and a(2)=1.
%C The a(n+2) represent the Kn13 and Kn23 sums of the square array of Delannoy numbers A008288. See A180662 for the definition of these knight and other chess sums.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,0,-1,1).
%F a(n) = a(n-1)+a(n-2)+a(n-3)+4*n-8 with a(0)=0, a(1)=0 and a(2)=1.
%F a(n) = a(n-1)+A001590(n+3)-2 with a(0)=0.
%F a(n) = sum(A008574(m)*A000073(n-m),m=0..n).
%F a(n+2) = add(A008288(n-k+2,k+2),k=0..floor(n/2)).
%F GF(x) = (x^2*(1+x)^2)/((1-x)^2*(1-x-x^2-x^3)).
%F Contribution from _Bruno Berselli_, Sep 23 2010: (Start)
%F a(n) = 2*a(n-1)-a(n-4)+4 for n>4.
%F a(n)-3*a(n-1)+2a(n-2)+a(n-4)-a(n-5) = 0 for n>4. (End)
%p nmax:=31: a(0):=0: a(1):=0: a(2):=1: for n from 3 to nmax do a(n):= a(n-1)+a(n-2)+a(n-3)+4*n-8 od: seq(a(n),n=0..nmax);
%t LinearRecurrence[{3,-2,0,-1,1},{0,0,1,5,14},40] (* _Harvey P. Dale_, Dec 15 2023 *)
%Y Cf. A000073 (Kn11 & Kn21), A089068 (Kn12 & Kn22), A180668 (Kn13 & Kn23), A180669 (Kn14 & Kn24), A180670 (Kn15 & Kn25).
%K easy,nonn
%O 0,4
%A _Johannes W. Meijer_, Sep 21 2010
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