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

0,4

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

Table of n, a(n) for n=0..29.

Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-3,3,-3,1).

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)).

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 *)

Crossrefs

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

Sequence in context: A061927 A292481 A051923 * A268262 A293101 A084899

Adjacent sequences: A180667 A180668 A180669 * A180671 A180672 A180673

Keyword

easy,nonn

Author

Johannes W. Meijer, Sep 21 2010

Status

approved