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

0,4

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).

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)

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 *)

Crossrefs

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

Cf. A000073, A005899, A008288.

Sequence in context: A006325 A053346 A227021 * A027964 A183957 A078501

Adjacent sequences: A180666 A180667 A180668 * A180670 A180671 A180672

Keyword

easy,nonn

Author

Johannes W. Meijer, Sep 21 2010

Status

approved