A092498 Expansion of g.f. (1 + x + 2*x^2)/((1 - x)^3*(1 - x^3)).

1, 4, 11, 23, 41, 67, 102, 147, 204, 274, 358, 458, 575, 710, 865, 1041, 1239, 1461, 1708, 1981, 2282, 2612, 2972, 3364, 3789, 4248, 4743, 5275, 5845, 6455, 7106, 7799, 8536, 9318, 10146, 11022, 11947, 12922, 13949, 15029, 16163, 17353, 18600, 19905, 21270

Offset: 0

Author

N. J. A. Sloane, Apr 05 2004

Keywords

• nonn
• easy

Comments

Arises from the Molien series for 4-dimensional group of structure S_3 X C_2 and order 12, which preserves the complete weight enumerators of even trace-Hermitian self-dual additive codes over GF(4). The Molien series is (1 + x^2 + 2*x^4)/((1 - x^2)^3 *(1 - x^6)).

From Philippe LALLOUET (philip.lallouet(AT)wanadoo.fr), Sep 24 2007: (Start)

Also arises when a pyramid is built row by row with squares of size 1.

At the first step, we put a single square on row 1. At the second, we put a square above the first one, on row 2, and a square on each of its sides on row 1. At each following step, we begin a new row with one square and add a square at each end of each of the previous rows. The term a(n) of the sequence is the total number of squares of any size which can be seen in the entire triangular array.

..........................

.....................|__|..

.....|__|.....|__||__|

||..||__||...||__||__||

The table below gives the number of squares by size, and the total number of squares (i.e., a(n)), for each row.

+-----------------------+

|size size size size |

n | 1 2 3 4 | a(n)

--+-----------------------+-----

1 | .1....................|....1

2 | .4....................|....4

3 | .9....2...............|...11

4 | 16....6....1..........|...23

5 | 25...12....4..........|...41

6 | 36...20....9....2.....|...67

(End)

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• G. Nebe, E. M. Rains, and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
• Index entries for Molien series
• Index entries for linear recurrences with constant coefficients, signature (3,-3,2,-3,3,-1).

Crossrefs

Cf. A014126.

Cf. A000969 (first differences). - R. J. Mathar, Jan 05 2009

Programs

Maple

A092498:=n->(4*n^3+21*n^2+35*n+18-6*floor((n+2)/3))/18; seq(A092498(n), n=0..50); # Wesley Ivan Hurt, Apr 19 2014

Mathematica

Table[(4*n^3 + 21*n^2 + 35*n + 18 - 6*Floor[(n + 2)/3])/18, {n, 0, 50}] (* Wesley Ivan Hurt, Apr 19 2014 *)
CoefficientList[Series[(1 + x + 2 x^2)/((1 - x)^3 (1 -x^3)), {x, 0, 40 }], x] (* Vincenzo Librandi, Apr 20 2014 *)
LinearRecurrence[{3,-3,2,-3,3,-1},{1,4,11,23,41,67},50] (* Harvey P. Dale, Jul 08 2017 *)

Formula

a(n) = (4*n^3+21*n^2+35*n+18-6*floor((n+2)/3))/18. - Luce ETIENNE, Apr 18 2014

a(n) = Sum_{j=0..floor(2*n/3)} ((4*n+5-6*j-(-1)^j)/4)*((4*n+3-6*j+(-1)^j)/4). - Luce ETIENNE, Oct 28 2014

E.g.f.: exp(-x/2)*(3*exp(3*x/2)*(2 + x)*(8 + 25*x + 4*x^2) + 6*cos(sqrt(3)*x/2) - 2*sqrt(3)*sin(sqrt(3)*x/2))/54. - Stefano Spezia, Apr 05 2023

Extensions

Edited by N. J. A. Sloane, May 15 2014