A180670 a(n) = a(n-1)+a(n-2)+a(n-3)+(8*n^3-48*n^2+112*n-96)/3 with a(0)=0, a(1)=0 and a(2)=1.

0, 0, 1, 9, 42, 140, 383, 925, 2056, 4316, 8705, 17069, 32810, 62192, 116743, 217673, 404000, 747496, 1380177, 2544865, 4688186, 8631620, 15886111, 29230725, 53776968, 98926372, 181971057, 334716197, 615660634, 1132400520

Offset: 0

Author

Johannes W. Meijer, Sep 21 2010

Keywords

• easy
• nonn

Comments

The a(n+2) represent the Kn15 and Kn25 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 (5,-9,7,-3,3,-3,1).

Crossrefs

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

Programs

Maple

nmax:=29: 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)+(8*n^3-48*n^2+112*n-96)/3 od: seq(a(n),n=0..nmax);

Mathematica

RecurrenceTable[{a[0]==a[1]==0,a[2]==1,a[n]==a[n-1]+a[n-2]+a[n-3]+(8n^3-48n^2+112n-96)/3},a,{n,30}] (* or *) LinearRecurrence[{5,-9,7,-3,3,-3,1},{0,0,1,9,42,140,383},30] (* Harvey P. Dale, Dec 04 2019 *)

Formula

a(n) = a(n-1)+a(n-2)+a(n-3)+(8*n^3-48*n^2+112*n-96)/3 with a(0)=0, a(1)=0 and a(2)=1.

a(n) = a(n-1)+A001590(n+7)-(12+4*n+4*n^2) with a(0)=0.

a(n) = sum(A008412(m)*A000073(n-m),m=0..n).

a(n+2) = add(A008288(n-k+4,k+4),k=0..floor(n/2)).

GF(x) = (x^2*(1+x)^4)/((1-x)^4*(1-x-x^2-x^3)).