A089068 a(n) = a(n-1)+a(n-2)+a(n-3)+2 with a(0)=0, a(1)=0 and a(2)=1.

0, 0, 1, 3, 6, 12, 23, 43, 80, 148, 273, 503, 926, 1704, 3135, 5767, 10608, 19512, 35889, 66011, 121414, 223316, 410743, 755475, 1389536, 2555756, 4700769, 8646063, 15902590, 29249424, 53798079, 98950095, 181997600, 334745776, 615693473

Offset: 0

Author

Roger L. Bagula, Dec 03 2003

Keywords

• nonn
• easy

Comments

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

Links

• Index entries for linear recurrences with constant coefficients, signature (2,0,0,-1).

Crossrefs

Cf. A000931, A000073, A077939, A113300, A001057, A006054, A033505.

Cf. A000073 (Kn11 & Kn21), A089068 (Kn12 & Kn22), A180668 (Kn13 & Kn23), A180669 (Kn14 & Kn24), A180670 (Kn15 & Kn25). - Johannes W. Meijer, Sep 21 2010

Programs

Mathematica

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 *)
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 *)

Formula

a(n) = A008937(n-2)+A008937(n-1). - Johannes W. Meijer, Sep 21 2010

a(n) = A018921(n-5)+A018921(n-4), n>4. - Johannes W. Meijer, Sep 21 2010

a(n) = A000073(n+2)-1. - R. J. Mathar, Sep 22 2010

From Johannes W. Meijer, Sep 22 2010: (Start)

a(n) = a(n-1)+A001590(n+1).

a(n) = Sum_{m=0..n} A040000(m)*A000073(n-m).

a(n+2) = Sum_{k=0..floor(n/2)} A008288(n-k+1,k+1).

G.f. = x^2*(1+x)/((1-x)*(1-x-x^2-x^3)). (End)

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

Extensions

Corrected and information added by Johannes W. Meijer, Sep 22 2010, Oct 22 2010

Definition based on arbitrarily set floating-point precision removed by R. J. Mathar, Sep 30 2010