A180669 a(n) = a(n-1)+a(n-2)+a(n-3)+4*n^2-16*n+18 with a(0)=0, a(1)=0 and a(2)=1.

0, 0, 1, 7, 26, 72, 171, 371, 760, 1500, 2889, 5475, 10266, 19116, 35435, 65495, 120832, 222664, 410017, 754671, 1388650, 2554784, 4699707, 8644907, 15901336, 29248068, 53796617, 98948523, 181995914, 334743972, 615691547

Offset: 0

Author

Johannes W. Meijer, Sep 21 2010

Keywords

• easy
• nonn

Comments

The a(n+2) represent the Kn14 and Kn24 sums of the square array of Delannoy numbers A008288. See A180662 for the definition of these knight and other chess sums.

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-5,2,-1,2,-1).

Crossrefs

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

Cf. A000073, A005899, A008288.

Programs

Maple

nmax:=30: 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-2)^2+2 od: seq(a(n),n=0..nmax);

Mathematica

nxt[{n_,a_,b_,c_}]:={n+1,b,c,a+b+c+4n(n-2)+6}; NestList[nxt,{2,0,0,1},30][[;;,2]] (* or *) LinearRecurrence[{4,-5,2,-1,2,-1},{0,0,1,7,26,72},40] (* Harvey P. Dale, Jul 13 2024 *)

Formula

a(n) = a(n-1)+a(n-2)+a(n-3)+4*(n-2)^2+2 with a(0)=0, a(1)=0 and a(2)=1.

a(n) = a(n-1)+A001590(n+5)-2-4*n with a(0)=0.

a(n) = Sum_{m=0..n} A005899(m)*A000073(n-m).

a(n+2) = Sum_{k=0..floor(n/2)} A008288(n-k+3,k+3).

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

From Bruno Berselli, Sep 23 2010: (Start)

a(n) = 3*a(n-1)-2a(n-2)-a(n-4)+a(n-5)+8 for n>4.

a(n)-4*a(n-1)+5a(n-2)-2*a(n-3)+a(n-4)-2*a(n-5)+a(n-6) = 0 for n>5. (End)