A180668 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.

0, 0, 1, 5, 14, 32, 67, 133, 256, 484, 905, 1681, 3110, 5740, 10579, 19481, 35856, 65976, 121377, 223277, 410702, 755432, 1389491, 2555709, 4700720, 8646012, 15902537, 29249369, 53798022, 98950036, 181997539, 334745713

Offset: 0

Author

Johannes W. Meijer, Sep 21 2010

Keywords

• easy
• nonn

Comments

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.

Links

• Index entries for linear recurrences with constant coefficients, signature (3,-2,0,-1,1).

Crossrefs

Cf. A000073 (Kn11 & Kn21), A089068 (Kn12 & Kn22), A180668 (Kn13 & Kn23), A180669 (Kn14 & Kn24), A180670 (Kn15 & Kn25).

Programs

Maple

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);

Mathematica

LinearRecurrence[{3,-2,0,-1,1},{0,0,1,5,14},40] (* Harvey P. Dale, Dec 15 2023 *)

Formula

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.

a(n) = a(n-1)+A001590(n+3)-2 with a(0)=0.

a(n) = sum(A008574(m)*A000073(n-m),m=0..n).

a(n+2) = add(A008288(n-k+2,k+2),k=0..floor(n/2)).

GF(x) = (x^2*(1+x)^2)/((1-x)^2*(1-x-x^2-x^3)).

Contribution from Bruno Berselli, Sep 23 2010: (Start)

a(n) = 2*a(n-1)-a(n-4)+4 for n>4.

a(n)-3*a(n-1)+2a(n-2)+a(n-4)-a(n-5) = 0 for n>4. (End)