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%I #33 Mar 13 2024 19:26:12
%S 0,0,1,3,9,16,24,36,49,63,81,100,120,144,169,195,225,256,288,324,361,
%T 399,441,484,528,576,625,675,729,784,840,900,961,1023,1089,1156,1224,
%U 1296,1369,1443,1521,1600,1680,1764,1849,1935,2025,2116,2208,2304,2401
%N Expansion of x^2*(1+x+4*x^2)/((1+x+x^2)*(1-x)^3).
%C From _Gerhard Kirchner_, Jan 20 2017: (Start)
%C According to the game "Mecanix":
%C In a triangular arrangement of wheel axles (n rows with 1, 2, ..., n axles), a connected set of unblocked gear wheels is installed such that the number of wheel quadruples forming half-hexagons is maximal.
%C a(n-1) is the maximum number.
%C Example:
%C Gear wheels (*) and free axles (·):
%C ·
%C * *
%C * * · *
%C · * · * * ·
%C * * · * * · * *
%C n=3 n=5
%C n=3: 1 half-hexagon, a(2)=1.
%C n=5: 3 half-hexagons and 1 full hexagon containing 6 half-hexagons -> a(4)=3+6*1=9.
%C See "Connected gear wheels" link.
%C Annotation: In such a configuration also the number of wheels is maximal. It is A007980(n). For n < 3, however, there is no half-hexagon. (End)
%C Floretion Algebra Multiplication Program, FAMP Code: 4tessumrokseq[A*B] with A = + .5'i + .5'j + .5'k + .5e and B = + .5i' + .5j' + .5k' + .5e; roktype: Y[15] = Y[15] + p; sumtype: Y[8] = (int)Y[6] - (int)Y[7] + Y[8] + sum (internal program code)
%H Gerhard Kirchner, <a href="/A109340/a109340.pdf">Connected gear wheels</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-2,1).
%F a(n+1) - a(n) = A047240(n);
%F a(n) + a(n+1) + a(n+2) = A056107(n);
%F a(n+2) - a(n+1) + a(n) = A105770(n).
%F a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5); a(0)=0, a(1)=0, a(2)=1, a(3)=3, a(4)=9. - _Harvey P. Dale_, Jun 24 2013
%F a(n) = (n-1)^2 - ((n+1) mod 3) mod 2, n >= 1. - _Gerhard Kirchner_, Jan 20 2017
%F E.g.f.: (exp(x)*(2 + 3*(x - 1)*x) - 2*exp(-x/2)*cos(sqrt(3)*x/2))/3. - _Stefano Spezia_, Dec 23 2022
%t CoefficientList[Series[x^2(1+x+4x^2)/((1+x+x^2)(1-x)^3),{x,0,50}],x] (* or *) LinearRecurrence[{2,-1,1,-2,1},{0,0,1,3,9},60] (* _Harvey P. Dale_, Jun 24 2013 *)
%Y Cf. A105770, A047240, A056107.
%K easy,nonn
%O 0,4
%A _Creighton Dement_, Aug 20 2005
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