A070333 Expansion of (1 + x)*(1 - x + x^2)/((1 - x)^4*(1 + x + x^2)).

1, 3, 6, 12, 21, 33, 50, 72, 99, 133, 174, 222, 279, 345, 420, 506, 603, 711, 832, 966, 1113, 1275, 1452, 1644, 1853, 2079, 2322, 2584, 2865, 3165, 3486, 3828, 4191, 4577, 4986, 5418, 5875, 6357, 6864, 7398, 7959, 8547, 9164, 9810, 10485

Offset: 0

Author

N. J. A. Sloane, May 11 2002

Keywords

• nonn
• easy

Comments

a(n) is the number of 3 X 3 matrices with nonnegative integer entries such that every row sum, column sum and the trace of the matrix is n. - Sharon Sela (sharonsela(AT)hotmail.com), May 20 2002

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,2,-3,3,-1).

Crossrefs

Cf. A059827, A001844, A002817.

Programs

Magma

[Round((2*n+3)*(n^2+3*n+6)/18): n in [0..50]]; // Vincenzo Librandi, Jun 25 2011

Maple

A049347 := proc(n) op(1+(n mod 3),[1,-1,0]) ; end proc:
A070333 := proc(n) 1+7*n/6+n^2/2+n^3/9+2*A049347(n-1)/9 ; end proc: # R. J. Mathar, Dec 03 2010

Mathematica

CoefficientList[ Series[(1 + x^3)/(1 - 3*x + 3*x^2 - 2*x^3 + 3*x^4 - 3*x^5 + x^6), {x, 0, 45}], x]

PARI

Vec((1+x)*(1-x+x^2)/((1-x)^4*(1+x+x^2))+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

Formula

a(n) = 1 + 7*n/6 + n^2/2 + n^3/9 + 2*A049347(n-1)/9. - R. J. Mathar, Dec 03 2010

From Mircea Merca, Dec 03 2010: (Start)

a(n) = round((2*n + 3)*(n^2 + 3*n + 6)/18).

a(n) = floor((n + 2)*(2*n^2 + 5*n + 11)/18).

a(n) = ceiling((n + 1)*(2*n^2 + 7*n + 14)/18).

a(n) = round((n + 1)*(2*n^2 + 7*n + 14)/18).

a(n) = a(n-3) + n^2 + 2 for n > 2. (End)

E.g.f.: exp(x)*(1 + x*(32 + x*(15 + 2*x))/18) + 4*exp(-x/2)*sin(sqrt(3)*x/2)/(9*sqrt(3)). - Stefano Spezia, Oct 28 2022