A109340 Expansion of x^2*(1+x+4*x^2)/((1+x+x^2)*(1-x)^3).

0, 0, 1, 3, 9, 16, 24, 36, 49, 63, 81, 100, 120, 144, 169, 195, 225, 256, 288, 324, 361, 399, 441, 484, 528, 576, 625, 675, 729, 784, 840, 900, 961, 1023, 1089, 1156, 1224, 1296, 1369, 1443, 1521, 1600, 1680, 1764, 1849, 1935, 2025, 2116, 2208, 2304, 2401

Offset: 0

Author

Creighton Dement, Aug 20 2005

Keywords

• easy
• nonn

Comments

From Gerhard Kirchner, Jan 20 2017: (Start)

According to the game "Mecanix":

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.

a(n-1) is the maximum number.

Example:

Gear wheels (*) and free axles (·):

·

* *

* * · *

· * · * * ·

* * · * * · * *

n=3 n=5

n=3: 1 half-hexagon, a(2)=1.

n=5: 3 half-hexagons and 1 full hexagon containing 6 half-hexagons -> a(4)=3+6*1=9.

See "Connected gear wheels" link.

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)

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)

Links

• Gerhard Kirchner, Connected gear wheels
• Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Crossrefs

Cf. A105770, A047240, A056107.

Programs

Mathematica

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 *)

Formula

a(n+1) - a(n) = A047240(n);

a(n) + a(n+1) + a(n+2) = A056107(n);

a(n+2) - a(n+1) + a(n) = A105770(n).

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

a(n) = (n-1)^2 - ((n+1) mod 3) mod 2, n >= 1. - Gerhard Kirchner, Jan 20 2017

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