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

0,4

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

Table of n, a(n) for n=0..31.

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

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)

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 *)

Crossrefs

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

Sequence in context: A266759 A139754 A036595 * A053209 A306192 A271993

Adjacent sequences: A180665 A180666 A180667 * A180669 A180670 A180671

Keyword

easy,nonn

Author

Johannes W. Meijer, Sep 21 2010

Status

approved