This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A089068 #39 Aug 04 2025 02:12:48
%S A089068 0,0,1,3,6,12,23,43,80,148,273,503,926,1704,3135,5767,10608,19512,
%T A089068 35889,66011,121414,223316,410743,755475,1389536,2555756,4700769,
%U A089068 8646063,15902590,29249424,53798079,98950095,181997600,334745776,615693473
%N A089068 a(n) = a(n-1)+a(n-2)+a(n-3)+2 with a(0)=0, a(1)=0 and a(2)=1.
%C A089068 The a(n+2) represent the Kn12 and Kn22 sums of the square array of Delannoy numbers A008288. See A180662 for the definition of these knight and other chess sums. - _Johannes W. Meijer_, Sep 21 2010
%H A089068 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,0,-1).
%F A089068 a(n) = A008937(n-2)+A008937(n-1). - _Johannes W. Meijer_, Sep 21 2010
%F A089068 a(n) = A018921(n-5)+A018921(n-4), n>4. - _Johannes W. Meijer_, Sep 21 2010
%F A089068 a(n) = A000073(n+2)-1. - _R. J. Mathar_, Sep 22 2010
%F A089068 From _Johannes W. Meijer_, Sep 22 2010: (Start)
%F A089068 a(n) = a(n-1)+A001590(n+1).
%F A089068 a(n) = Sum_{m=0..n} A040000(m)*A000073(n-m).
%F A089068 a(n+2) = Sum_{k=0..floor(n/2)} A008288(n-k+1,k+1).
%F A089068 G.f. = x^2*(1+x)/((1-x)*(1-x-x^2-x^3)). (End)
%F A089068 a(n) = 2*a(n-1)-a(n-4), a(0)=0, a(1)=0, a(2)=1, a(3)=3. - _Bruno Berselli_, Sep 23 2010
%t A089068 Join[{a=0,b=0,c=1},Table[d=a+b+c+2;a=b;b=c;c=d,{n,50}]] (* _Vladimir Joseph Stephan Orlovsky_, Apr 19 2011 *)
%t A089068 RecurrenceTable[{a[0]==a[1]==0,a[2]==1,a[n]==a[n-1]+a[n-2]+a[n-3]+2}, a[n],{n,40}] (* or *) LinearRecurrence[{2,0,0,-1},{0,0,1,3},40] (* _Harvey P. Dale_, Sep 19 2011 *)
%Y A089068 Cf. A000931, A000073, A077939, A113300, A001057, A006054, A033505.
%Y A089068 Cf. A000073 (Kn11 & Kn21), A089068 (Kn12 & Kn22), A180668 (Kn13 & Kn23), A180669 (Kn14 & Kn24), A180670 (Kn15 & Kn25). - _Johannes W. Meijer_, Sep 21 2010
%K A089068 nonn,easy
%O A089068 0,4
%A A089068 _Roger L. Bagula_, Dec 03 2003
%E A089068 Corrected and information added by _Johannes W. Meijer_, Sep 22 2010, Oct 22 2010
%E A089068 Definition based on arbitrarily set floating-point precision removed by _R. J. Mathar_, Sep 30 2010