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Expansion of g.f. (1 + x + 2*x^2)/((1 - x)^3*(1 - x^3)).
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%I #55 Apr 05 2023 18:51:02

%S 1,4,11,23,41,67,102,147,204,274,358,458,575,710,865,1041,1239,1461,

%T 1708,1981,2282,2612,2972,3364,3789,4248,4743,5275,5845,6455,7106,

%U 7799,8536,9318,10146,11022,11947,12922,13949,15029,16163,17353,18600,19905,21270

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

%C 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)).

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

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

%C 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.

%C ..........................__

%C ..........__...........__|__|__..

%C .__....__|__|__.....__|__|__|__|__

%C |__|..|__|__|__|...|__|__|__|__|__|

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

%C +-----------------------+

%C |size size size size |

%C n | 1 2 3 4 | a(n)

%C --+-----------------------+-----

%C 1 | .1....................|....1

%C 2 | .4....................|....4

%C 3 | .9....2...............|...11

%C 4 | 16....6....1..........|...23

%C 5 | 25...12....4..........|...41

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

%C (End)

%H Vincenzo Librandi, <a href="/A092498/b092498.txt">Table of n, a(n) for n = 0..1000</a>

%H G. Nebe, E. M. Rains, and N. J. A. Sloane, <a href="http://neilsloane.com/doc/cliff2.html">Self-Dual Codes and Invariant Theory</a>, Springer, Berlin, 2006.

%H <a href="/index/Mo#Molien">Index entries for Molien series</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,2,-3,3,-1).

%F a(n) = (4*n^3+21*n^2+35*n+18-6*floor((n+2)/3))/18. - _Luce ETIENNE_, Apr 18 2014

%F 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

%F 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

%p 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

%t 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 *)

%t CoefficientList[Series[(1 + x + 2 x^2)/((1 - x)^3 (1 -x^3)), {x, 0, 40 }], x] (* _Vincenzo Librandi_, Apr 20 2014 *)

%t LinearRecurrence[{3,-3,2,-3,3,-1},{1,4,11,23,41,67},50] (* _Harvey P. Dale_, Jul 08 2017 *)

%Y Cf. A014126.

%Y Cf. A000969 (first differences). - _R. J. Mathar_, Jan 05 2009

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Apr 05 2004

%E Edited by _N. J. A. Sloane_, May 15 2014